R.D. Sharma solutions for Class 12 Mathematics chapter 24 - Scalar Or Dot Product

Represent the following graphically:
(i) a displacement of 40 km, 30° east of north
(ii) a displacement of 50 km south-east
(iii) a displacement of 70 km, 40° north of west.

Classify the following measures as scalars and vectors:
(i) 15 kg
(ii) 20 kg weight
(iii) 45°
(iv) 10 meters south-east
(v) 50 m/sec²

Classify the following as scalars and vector quantities:
(i) Time period
(ii) Distance
(iii) displacement
(iv) Force
(v) Work
(vi) Velocity
(vii) Acceleration

Answer the following as true or false:
\[\vec{a}\] and \[\vec{a}\]  are collinear.

True

False

Answer the following as true or false:
Two collinear vectors are always equal in magnitude.

true

False

Answer the following as true or false:
Zero vector is unique.

true 

false

Answer the following as true or false:
Two vectors having same magnitude are collinear.

true

false

Answer the following as true or false:
Two collinear vectors having the same magnitude are equal.

true

false

If P, Q and R are three collinear points such that \[\vec{PQ} = \vec{a}\] and \[\vec{PQ} = \vec{a}\]. Find the vector \[\vec{PR}\]

Give a condition that three vectors \[\vec{a}\], \[\vec{b}\] and \[\vec{b}\]  form the three sides of a triangle. What are the other possibilities?

If \[\vec{a}\] and \[\vec{b}\] are two non-collinear vectors having the same initial point. What are the vectors represented by \[\vec{a}\] + \[\vec{b}\]  and \[\vec{a}\] − \[\vec{b}\].

If \[\vec{a}\] is a vector and m is a scalar such that m \[\vec{a}\] = \[\vec{0}\], then what are the alternatives for m and \[\vec{a}\] ?

If \[\vec{a,} \vec{b}\] are two vectors, then write the truth value of the following statement: 
 \[\vec{a} = - \vec{b} \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right|\]

If \[\vec{a,} \vec{b}\] are two vectors, then write the truth value of the following statement: 
\[\vec{a} = - \vec{b} \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right|\]

If \[\vec{a,} \vec{b}\] are two vectors, then write the truth value of the following statement: 
\[\left| \vec{a} \right| = \left| \vec{b} \right| \Rightarrow \vec{a} = \vec{b}\]

ABCD is a quadrilateral. Find the sum the vectors \[\vec{BA} , \vec{BC} , \vec{CD}\] and \[\vec{DA}\]

ABCDE is a pentagon, prove that
\[\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} + \vec{EA} = \vec{0}\]

ABCDE is a pentagon, prove that 
\[\vec{AB} + \vec{AE} + \vec{BC} + \vec{DC} + \vec{ED} + \vec{AC} = 3 \vec{AC}\]

If P is a point and ABCD is a quadrilateral and \[\vec{AP} + \vec{PB} + \vec{PD} = \vec{PC}\], show that ABCD is a parallelogram.

Five forces \[\vec{AB,} \vec{AC,} \vec{AD,} \vec{AE}\] and \[\vec{AF}\] act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6 \[\vec{AO,}\] where O is the centre of hexagon.

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors \[\vec{OP} = 2 \vec{a} + \vec{b}\] and \[\vec{OQ} = \vec{a} - 2 \vec{b}\], respectively in the ratio 1 : 2 internally and externally. 

Let \[\vec{a,} \vec{b,} \vec{c,} \vec{d}\] be the position vectors of the four distinct points A, B, C, D. If \[\vec{b} - \vec{a} = \vec{c} - \vec{d}\], then show that ABCD is a parallelogram.

If \[\vec{a,} \vec{b}\] are the position vectors of A, B respectively, find the position vector of a point C in AB produced such that AC = 3 AB and that a point D in BA produced such that BD = 2BA.

Show that the four points A, B, C, D with position vectors \[\vec{a,} \vec{b,} \vec{c,} \vec{d}\] respectively such that \[3 \vec{a} - 2 \vec{b} + 5 \vec{c} - 6 \vec{d} = 0,\] are coplanar. Also, find the position vector of the point of intersection of the line segments AC and BD.

Show that the four points P, Q, R, S with position vectors \[\vec{p}\], \[\vec{q}\], \[\vec{r}\], \[\vec{s}\] respectively such that 5 \[\vec{p}\] − 2 \[\vec{q}\] + 6 \[\vec{r}\] − 9 \[\vec{s}\] \[\vec{0}\], are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.

The vertices A, B, C of triangle ABC have respectively position vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\]  with respect to a given origin O. Show that the point D where the bisector of ∠ A meets BC has position vector \[\vec{d} = \frac{\beta \vec{b} + \gamma \vec{c}}{\beta + \gamma},\text{ where }\beta = \left| \vec{c} - \vec{a} \right| \text{ and, }\gamma = \left| \vec{a} - \vec{b} \right|\]
Hence, deduce that the incentre I has position vector
\[\frac{\alpha \vec{a} + \beta \vec{b} + \gamma \vec{c}}{\alpha + \beta + \gamma},\text{ where }\alpha = \left| \vec{b} - \vec{c} \right|\]

If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that \[\vec{OA} + \vec{OB} + \vec{OC} = \vec{OD} + \vec{OE} + \vec{OF}\]

ABCD is a parallelogram and P is the point of intersection of its diagonals. If O is the origin of reference, show that
\[\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = 4 \vec{OP}\]

ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD; BC and AD. Show that\[\vec{PA} + \vec{PB} + \vec{PC} + \vec{PD} = 4 \vec{PQ}\], where P is any point.

If the position vector of a point (−4, −3) be \[\vec{a,}\] find \[\left| \vec{a} \right|\]

If the position vector \[\vec{a}\] of a point (12, n) is such that \[\left| \vec{a} \right|\] = 13, find the value (s) of n.

Find a vector of magnitude 4 units which is parallel to the vector \[\sqrt{3} \hat{i} + \hat{j}\]

Express \[\vec{AB}\]  in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (4, −1), B (1, 3)
Find \[\left| \vec{A} B \right|\] in each case.

Express \[\vec{AB}\]  in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (−6, 3), B (−2, −5)
Find \[\left| \vec{A} B \right|\] in each case.

Find the coordinates of the tip of the position vector which is equivalent to \[\vec{A} B\], where the coordinates of A and B are (−1, 3) and (−2, 1) respectively.

If the position vectors of the points A (3, 4), B (5, −6) and C (4, −1) are \[\vec{a,}\] \[\vec{b,}\] \[\vec{c}\] respectively, compute \[\vec{a} + 2 \vec{b} - 3 \vec{c}\].

If \[\vec{a}\] be the position vector whose tip is (5, −3), find the coordinates of a point B such that \[\vec{AB} =\] \[\vec{a}\], the coordinates of A being (4, −1).

Show that the points 2 \[\hat{i}, -    \hat{i}-4 \]
\[\hat{j}\] and \[\hat{i}+4\]  form an isosceles triangle.

Find a unit vector parallel to the vector \[\hat{i} + \sqrt{3} \hat{j}\]

The position vectors of points A, B and C  are \[\lambda \hat{i} +\] 3 \[\hat{j}\],12\[\hat{i} + \mu\] \[\hat{j}\] and 11\[\hat{i} -\] 3 \[\hat{j}\] respectively. If C divides the line segment joining A and B in the ratio 3:1, find the values of \[\lambda\] and \[\lambda\]

Find the magnitude of the vector \[\vec{a} = 2 \hat{i} + 3 \hat{j} - 6 \hat{k} .\]

Find the unit vector in the direction of \[3 \hat{i} + 4 \hat{j} - 12 \hat{k} .\]

Find a unit vector in the direction of the resultant of the vectors
\[\hat{i} - \hat{j} + 3 \hat{k} , 2 \hat{i} + \hat{j} - 2 \hat{k} \text{ and }\hat{i} + 2 \hat{j} - 2 \hat{k} .\]

The adjacent sides of a parallelogram are represented by the vectors \[\vec{a} = \hat{i} + \hat{j} - \hat{k}\text{ and }\vec{b} = - 2 \hat{i} + \hat{j} + 2 \hat{k} .\]
Find unit vectors parallel to the diagonals of the parallelogram.

If \[\vec{PQ} = 3 \hat{i} + 2 \hat{j} - \hat{k}\] and the coordinates of P are (1, −1, 2), find the coordinates of Q.

Prove that the points \[\hat{i} - \hat{j} , 4 \hat{i} + 3 \hat{j} + \hat{k} \text{ and }2 \hat{i} - 4 \hat{j} + 5 \hat{k}\] are the vertices of a right-angled triangle.

If the vertices of a triangle are the points with position vectors \[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} , b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} , c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k} ,\]
what are the vectors determined by its sides? Find the length of these vectors.

Find the position vector of a point R which divides the line segment joining points \[P \left( \hat{i} + 2 \hat{j} + \hat{k} \right) \text{ and Q }\left( - \hat{i} + \hat{j} + \hat{k} \right)\] internally.

Find the position vector of the mid-point of the vector joining the points \[P \left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right)\text{ and Q }\left( 4 \hat{i} + \hat{j} - 2 \hat{k} \right) .\]

Find the position vector of the mid-point of the vector joining the points \[P \left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right)\text{ and Q }\left( 4 \hat{i} + \hat{j} - 2 \hat{k} \right) .\]

Find the unit vector in the direction of vector \[\vec{PQ} ,\]

Show that the points \[A \left( 2 \hat{i} - \hat{j} + \hat{k} \right), B \left( \hat{i} - 3 \hat{j} - 5 \hat{k} \right), C \left( 3 \hat{i} - 4 \hat{j} - 4 \hat{k} \right)\] are the vertices of a right angled triangle.

Find the value of x for which \[x \left( \hat{i} + \hat{j} + \hat{k} \right)\] is a unit vector.

If \[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k} \text{ and }\vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,\] find a unit vector parallel to \[2 \vec{a} - \vec{b} + 3 \vec{c .}\] 

If \[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 4 \hat{i} - 2 \hat{j} + 3 \hat{k} and \vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,\] find a vector of magnitude 6 units which is parallel to the vector \[2 \vec{a} - \vec{b} + 3 \vec{c .}\]

Find a vector of magnitude of 5 units parallel to the resultant of the vectors \[\vec{a} = 2 \hat{i} + 3 \hat{j} - \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} +\widehat{k} .\]

The two vectors \[\hat{j} + \hat{k}\] and \[3 \hat{i} - \hat{j} + 4 \hat{k}\] represents the sides \[\vec{AB}\] and \[\vec{AC}\] respectively of a triangle ABC. Find the length of the median through A.

Show that the points A, B, C with position vectors \[\vec{a} - 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + 3 \vec{b} - 4 \vec{c}\] and \[- 7 \vec{b} + 10 \vec{c}\] are collinear.

If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are non-coplanar vectors, prove that the points having the following position vectors are collinear: \[\vec{a,} \vec{b,} 3 \vec{a} - 2 \vec{b}\]

If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are non-coplanar vectors, prove that the points having the following position vectors are collinear: \[\vec{a} + \vec{b} + \vec{c} , 4 \vec{a} + 3 \vec{b} , 10 \vec{a} + 7 \vec{b} - 2 \vec{c}\]

Prove that the points having position vectors \[\hat{i} + 2 \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k} , - 3 \hat{i} - 2 \hat{i} - 5 \hat{k}\] are collinear.

If the points with position vectors \[10 \hat{i} + 3 \hat{j} , 12 \hat{i} - 5 \hat{j}\text{ and a }\hat{i} + 11 \hat{j}\] are collinear, find the value of a.

If \[\vec{a,} \vec{b}\] are two non-collinear vectors, prove that the points with position vectors \[\vec{a,} \vec{b}\] are two non-collinear vectors, prove that the points with position vectors \[\vec{a} + \vec{b,} \vec{a} - \vec{b}\text{ and }\vec{a} + \lambda \vec{b}\] are collinear for all real values of λ.

If \[\vec{AO} + \vec{OB} = \vec{BO} + \vec{OC} ,\] prove that A, B, C are collinear points.

Show that the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k}\text{ and }- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\] are collinear.

If the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j}\] and \[\vec{b} = - 6 \hat{i} + m \hat{j}\] are collinear, find the value of m.

Show that the points whose position vectors are as given below are collinear:
\[2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k} \text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}\]

Show that the points whose position vectors are as given below are collinear: \[3 \hat{i} - 2 \hat{j} + 4 \hat{k}, \hat{i} + \hat{j} + \hat{k}\text{ and }- \hat{i} + 4 \hat{j} - 2 \hat{k}\]

Using vector method, prove that the following points are collinear:
A (6, −7, −1), B (2, −3, 1) and C (4, −5, 0)

Using vector method, prove that the following points are collinear:
A (2, −1, 3), B (4, 3, 1) and C (3, 1, 2)

Using vector method, prove that the following points are collinear:
A (1, 2, 7), B (2, 6, 3) and C (3, 10, −1)

Using vector method, prove that the following points are collinear:
A (−3, −2, −5), B (1, 2, 3) and C (3, 4, 7)

If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are non-zero, non-coplanar vectors, prove that the following vectors are coplanar:
(1) \[5 \vec{a} + 6 \vec{b} + 7 \vec{c,} 7 \vec{a} - 8 \vec{b} + 9 \vec{c}\text{ and }3 \vec{a} + 20 \vec{b} + 5 \vec{c}\]

Show that the four points having position vectors
\[6 \hat{i} - 7 \hat{j} , 16 \hat{i} - 19 \hat{j} - 4 \hat{k} , 3 \hat{j} - 6 \hat{k} , 2 \hat{i} - 5 \hat{j} + 10 \hat{k}\] are coplanar.

Prove that the following vectors are coplanar:
\[2 \hat{i} - \hat{j} + \hat{k} , \hat{i} - 3 \hat{j} - 5 \hat{k} \text{ and }3 \hat{i} - 4 \hat{j} - 4 \hat{k}\]

Prove that the following vectors are coplanar:
\[\hat{i} + \hat{j} + \hat{k} , 2 \hat{i} + 3 \hat{j} - \hat{k}\text{ and }- \hat{i} - 2 \hat{j} + 2 \hat{k}\]

Prove that the following vectors are non-coplanar:

Prove that the following vectors are non-coplanar:

If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[2 \vec{a} - \vec{b} + 3 \vec{c} , \vec{a} + \vec{b} - 2 \vec{c}\text{ and }\vec{a} + \vec{b} - 3 \vec{c}\]

If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[\vec{a} + 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + \vec{b} + 3 \vec{c}\text{ and }\vec{a} + \vec{b} + \vec{c}\]

Show that the vectors \[\vec{a,} \vec{b,} \vec{c}\] given by \[\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}\text{ and }\vec{c} = \hat{i} + \hat{j} + \hat{k}\]  are non-coplanar.
Express vector \[\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}\text{ and }\vec{c} = \hat{i} + \hat{j} + \hat{k}\]  as a linear combination of the vectors \[\vec{a,} \vec{b}\text{ and }\vec{c} .\]

Prove that a necessary and sufficient condition for three vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\]  to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]

Show that the four points A, B, C and D with position vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\], \[\vec{d}\] respectively are coplanar if and only if \[3 \vec{a} - 2 \vec{b} + \vec{c} - 2 \vec{d} = \vec{0} .\]

A vector makes an angle of \[\frac{\pi}{4}\] with each of x-axis and y-axis. Find the angle made by it with the z-axis.

A vector \[\vec{r}\] is inclined at equal acute angles to x-axis, y-axis and z-axis.
If |\[\vec{r}\]| = 6 units, find \[\vec{r}\].

A vector \[\vec{r}\] is inclined to -axis at 45° and y-axis at 60°.
If \[\vec{r}\] = 8 units, find \[\vec{r}\].

Find the direction cosines of the following vectors:
\[2 \hat{i} + 2 \hat{j} - \hat{k}\]

Find the direction cosines of the following vectors:
\[6 \hat{i} - 2 \hat{j} - 3 \hat{k}\]

Find the direction cosines of the following vectors:
\[3 \hat{i} - 4 \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[\hat{i} - \hat{j} + \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[\hat{j} - \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[4 \hat{i} + 8 \hat{j} + \hat{k}\]

Show that the vector \[\hat{i} + \hat{j} + \hat{k}\] is equally inclined with the axes OX, OY and OZ.

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are \[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} .\]

If a unit vector \[\vec{a}\] makes an angle \[\frac{\pi}{3}\] with \[\hat{i} , \frac{\pi}{4}\] with \[\hat{j}\]  and an acute angle θ with \[\hat{k}\], then find θ and hence, the components of \[\vec{a}\].

Find a vector \[\vec{r}\] of magnitude \[3\sqrt{2}\] units which makes an angle of \[\frac{\pi}{4}\] and \[\frac{\pi}{4}\] with y and z-axes respectively. 

A vector \[\vec{r}\] is inclined at equal angles to the three axes. If the magnitude of \[\vec{r}\] is \[2\sqrt{3}\], find \[\vec{r}\].

Write \[\vec{PQ} + \vec{RP} + \vec{QR}\] in the simplified form.

If \[\vec{a}\] and \[\vec{b}\] are two non-collinear vectors such that \[x \vec{a} + y \vec{b} = \vec{0} ,\] then write the values of x and y.

If \[\vec{a}\] and \[\vec{b}\] represent two adjacent sides of a parallelogram, then write vectors representing its diagonals.

If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] represent the sides of a triangle taken in order, then write the value of \[\vec{a} + \vec{b} + \vec{c} .\]

If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are position vectors of the vertices A, B and C respectively, of a triangle ABC, write the value of \[\vec{AB} + \vec{BC} + \vec{CA} .\]

If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\]  are position vectors of the points A, B and C respectively, write the value of \[\vec{AB} + \vec{BC} + \vec{AC} .\]

If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are the position vectors of the vertices of a triangle, then write the position vector of its centroid.

If G denotes the centroid of ∆ABC, then write the value of \[\vec{GA} + \vec{GB} + \vec{GC} .\]

If \[\vec{a}\] and \[\vec{b}\] denote the position vectors of points A and B respectively and C is a point on ABsuch that 3AC = 2AB, then write the position vector of C.

If D is the mid-point of side BC of a triangle ABC such that \[\vec{AB} + \vec{AC} = \lambda \vec{AD} ,\] write the value of λ.

If D, E, F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC, write the value of \[\vec{AD} + \vec{BE} + \vec{CF} .\]

If \[\vec{a}\] is a non-zero vector of modulus a and m is a non-zero scalar such that m \[\vec{a}\] is a unit vector, write the value of m.

If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then write the value of \[\vec{a} + \vec{b} + \vec{c} .\]

Write the position vector of a point dividing the line segment joining points A and B with position vectors \[\vec{a}\] and \[\vec{b}\] externally in the ratio 1 : 4, where \[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \text{ and }\vec{b} = - \hat{i} + \hat{j} + \hat{k} .\]

Write the direction cosines of the vector \[\vec{r} = 6 \hat{i} - 2 \hat{j} + 3 \hat{k} .\]

If \[\vec{a} = \hat{i} + \hat{j} , \vec{b} = \hat{j} + \hat{k} \text{ and }\vec{c} = \hat{k} + \hat{i} ,\] write unit vectors parallel to \[\vec{a} + \vec{b} - 2 \vec{c} .\]

If \[\vec{a} = \hat{i} + 2 \hat{j} , \vec{b} = \hat{j} + 2 \hat{k} ,\] write a unit vector along the vector \[3 \vec{a} - 2 \vec{b} .\]

Write the position vector of a point dividing the line segment joining points having position vectors \[\hat{i} + \hat{j} - 2 \hat{k} \text{ and }2 \hat{i} - \hat{j} + 3 \hat{k}\] externally in the ratio 2:3.

If \[\vec{a} = \hat{i} + \hat{j} , \vec{b} = \hat{j} + \hat{k} , \vec{c} = \hat{k} + \hat{i}\], find the unit vector in the direction of \[\vec{a} + \vec{b} + \vec{c}\].

A unit vector \[\vec{r}\] makes angles \[\frac{\pi}{3}\] and \[\frac{\pi}{2}\] with \[\hat{j}\text{ and }\hat{k}\]  respectively and an acute angle θ with \[\hat{i}\]. Find θ.

Write a unit vector in the direction of \[\vec{a} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} .\]

If \[\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k} \text{ and }\vec{b} = 2 \hat{i} + 4 \hat{j} + 9 \hat{k} ,\]  find a unit vector parallel to \[\vec{a} + \vec{b}\].

Write a unit vector in the direction of \[\vec{b} = 2 \hat{i} + \hat{j} + 2 \hat{k}\].

Find the position vector of the mid-point of the line segment AB, where A is the point (3, 4, −2) and B is the point (1, 2, 4).

Find a vector in the direction of \[\vec{a} = 2 \hat{i} - \hat{j} + 2 \hat{k} ,\] which has magnitude of 6 units.

What is the cosine of the angle which the vector \[\sqrt{2} \hat{i} + \hat{j} + \hat{k}\] makes with y-axis?

Write a vector in the direction of vector \[5 \hat{i} - \hat{j} + 2 \hat{k}\] which has magnitude of 8 unit.

Write the direction cosines of the vector \[\hat{i} + 2 \hat{j} + 3 \hat{k}\].

Find a unit vector in the direction of \[\vec{a} = 2 \hat{i} - 3 \hat{j} + 6 \hat{k}\].

For what value of 'a' the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k} \text{ and }a \hat{i} + 6 \hat{j} - 8 \hat{k}\]  are collinear?

Write the direction cosines of the vectors \[- 2 \hat{i} + \hat{j} - 5 \hat{k}\].

Find the sum of the following vectors: \[\vec{a} = \hat{i} - 2 \hat{j} , \vec{b} = 2 \hat{i} - 3 \hat{j} , \vec{c} = 2 \hat{i} + 3 \hat{k} .\]

Find a unit vector in the direction of the vector \[\vec{a} = 3 \hat{i} - 2 \hat{j} + 6 \hat{k}\].

If \[\vec{a} = x \hat{i} + 2 \hat{j} - z \hat{k}\text{ and }\vec{b} = 3 \hat{i} - y \hat{j} + \hat{k}\]  are two equal vectors, then write the value of x + y + z.

Write a unit vector in the direction of the sum of the vectors \[\vec{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}\] and \[\vec{b} = 2 \hat{i} + \hat{j} - 7 \hat{k}\].

Find the value of 'p' for which the vectors \[3 \hat{i} + 2 \hat{j} + 9 \hat{k}\] and \[\hat{i} - 2p \hat{j} + 3 \hat{k}\] are parallel.

Find a vector \[\vec{a}\] of magnitude \[5\sqrt{2}\], making an angle of \[\frac{\pi}{4}\] with x-axis, \[\frac{\pi}{2}\] with y-axis and an acute angle θ with z-axis. 

Write a unit vector in the direction of \[\vec{PQ}\], where P and Q are the points (1, 3, 0) and (4, 5, 6) respectively.

Find a vector in the direction of vector \[2 \hat{i} - 3 \hat{j} + 6 \hat{k}\] which has magnitude 21 units.

If \[\left| \vec{a} \right| = 4\] and \[- 3 \leq \lambda \leq 2\], then write the range of \[\left| \lambda \vec{a} \right|\].

In a triangle OAC, if B is the mid-point of side AC and \[\vec{OA} = \vec{a} , \vec{OB} = \vec{b}\], then what is \[\vec{OC}\].

Write the position vector of the point which divides the join of points with position vectors \[3 \vec{a} - 2 \vec{b}\text{ and }2 \vec{a} + 3 \vec{b}\] in the ratio 2 : 1.

If in a ∆ABC, A = (0, 0), B = (3, 3 \[\sqrt{3}\]), C = (−3\[\sqrt{3}\], 3), then the vector of magnitude 2 \[\sqrt{2}\] units directed along AO, where O is the circumcentre of ∆ABC is 

none of these

If \[\vec{a} , \vec{b}\] are the vectors forming consecutive sides of a regular hexagon ABCDEF, then the vector representing side CD is 

Forces 3 O \[\vec{A}\], 5 O \[\vec{B}\] act along OA and OB. If their resultant passes through C on AB, then 

C is a mid-point of AB

C divides AB in the ratio 2 : 1

3 AC = 5 CB

2 AC = 3 CB

If \[\vec{a} , \vec{b} , \vec{c}\] are three non-zero vectors, no two of which are collinear and the vector \[\vec{a} + \vec{b}\] is collinear with \[\vec{c} , \vec{b} + \vec{c}\] is collinear with \[\vec{a} ,\] then \[\vec{a} + \vec{b} + \vec{c} =\]

none of these

If points A (60 \[\hat{i}\] + 3 \[\hat{j}\]), B (40 \[\hat{i}\] − 8 \[\hat{j}\]) and C (a \[\hat{i}\] − 52 \[\hat{j}\]) are collinear, then a is equal to

40

−40

20

 −20

If G is the intersection of diagonals of a parallelogram ABCD and O is any point, then \[O \vec{A} + O \vec{B} + O \vec{C} + O \vec{D} =\] 

The vector cos α cos β \[\hat{i}\] + cos α sin β \[\hat{j}\] + sin α \[\hat{k}\] is a

null vector

unit vector

constant vector

none of these

In a regular hexagon ABCDEF, A \[\vec{B}\] = a, B \[\vec{C}\] = \[\vec{b}\text{ and }\vec{CD} = \vec{c}\].
Then, \[\vec{AE}\] =

\[2 \vec{a} + \vec{b} + \vec{c}\]

\[\vec{a} + 2 \vec{b} + 2 \vec{c}\]

The vector equation of the plane passing through \[\vec{a} , \vec{b} , \vec{c} ,\text{ is }\vec{r} = \alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} ,\] provided that

α + β + γ = 0

α + β + γ =1

α + β = γ

α² + β² + γ² = 1

If O and O' are circumcentre and orthocentre of ∆ ABC, then \[\vec{OA} + \vec{OB} + \vec{OC}\] equals 

2\[\vec{OO}\]

If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] and \[\vec{d}\] are the position vectors of points A, B, C, D such that no three of them are collinear and \[\vec{a} + \vec{c} = \vec{b} + \vec{d} ,\] then ABCD is a

 rhombus

rectangle

square

parallelogram

Let G be the centroid of ∆ ABC. If \[\vec{AB} = \vec{a,} \vec{AC} = \vec{b,}\] then the bisector \[\vec{AG} ,\] in terms of \[\vec{a}\text{ and }\vec{b}\] is

\[\frac{2}{3}\left( \vec{a} + \vec{b} \right)\]

If ABCDEF is a regular hexagon, then \[\vec{AD} + \vec{EB} + \vec{FC}\] equals

\[2 \vec{AB}\]

\[3 \vec{AB}\]

The position vectors of the points A, B, C are \[2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k}\text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}\] respectively.
These points

 form an isosceles triangle

form a right triangle

are collinear

form a scalene triangle

If three points A, B and C have position vectors \[\hat{i} + x \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k}\text{ and }y \hat{i} - 2 \hat{j} - 5 \hat{k}\] respectively are collinear, then (x, y) =

 (2, −3)

(−2, 3)

 (−2, −3)

(2, 3)

ABCD is a parallelogram with AC and BD as diagonals.
Then, \[\vec{AC} - \vec{BD} =\] 

\[4 \vec{AB}\]

If OACB is a parallelogram with \[\vec{OC} = \vec{a}\text{ and }\vec{AB} = \vec{b} ,\] then \[\vec{OA} =\]

If \[\vec{a}\text{ and }\vec{b}\] are two collinear vectors, then which of the following are incorrect?

\[\vec{b} = \lambda \vec{a}\] for some scalar λ

the respective components of \[\vec{a}\text{ and }\vec{b}\] are proportional

both the vectors \[\vec{a}\text{ and }\vec{b}\] have the same direction but different magnitudes

Find \[\vec{a} \cdot \vec{b}\] when

\[\vec{a} =\hat{i} - 2\hat{j} + \hat{k}\text{ and } \vec{b} = 4 \hat{i} - 4\hat{j} + 7 \hat{k}\]

Find \[\vec{a} \cdot \vec{b}\] when

\[\vec{a} = \hat{j} + 2 \hat{k}  \text{ and } \vec{b} = 2 \hat{i} + \hat{k}\]

Find \[\vec{a} \cdot \vec{b}\] when 

\[\vec{a} = \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + 3 \hat{j} - 2 \hat{k}\]

For what value of λ are the vectors \[\vec{a} \text{ and  }\vec{b}\] perpendicular to each other if \[\vec{a} = \lambda \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{b} = 4\hat{i} - 9 \hat{j} + 2\hat{k}\] 

For what value of λ are the vectors \[\vec{a} and \vec{b}\] perpendicular to each other if  

\[\vec{a} = \lambda \hat{i} + 2\hat{j} + \hat{k} \text{ and } \vec{b} = 5\hat{i} - 9 \hat{j} + 2\hat{k}\]

For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if \[\vec{a} = \lambda \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{b} = 5 \hat{i} - 9 \hat{j} + 2\hat{k}\]

For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if  

\[\vec{a} = \lambda \hat{i} + 3 \hat{j} + 2 \hat{k}\text { and } \vec{b} = \hat{i} - \hat{j} + 3 \hat{k}\]

If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} \right| = 4, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 6\] find the angle between \[\vec{a} \text{ and } \vec{b} .\]

\[\text{ If } \vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + 2\hat{k} , \text{find} \left( \vec{a} - 2 \vec{b} \right) \cdot \left( \vec{a} + \vec{b} \right) .\]

Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] where \[\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = \hat{j} + \hat{k}\]

Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] \[\vec{a} = 3\hat{i} - 2\hat{j} - 6\hat{k} \text{ and } \vec{b} = 4 \hat{i} - \hat{j} + 8 \hat{k}\]

Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\]  \[\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \text{ and } \vec{b} = 4\hat{i} + 4 \hat{j} - 2\hat{k}\]

Find the angle between the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - 2 \hat{k}\]

Find the angle between the vectors \[\vec{a} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k}\]

Find the angles which the vector \[\vec{a} = \hat{i} -\hat {j} + \sqrt{2} \hat{k}\] makes with the coordinate axes.

Dot product of a vector with \[\hat{i} + \hat{j} - 3\hat{k} , \hat{i} + 3\hat{j} - 2 \hat{k} \text{ and } 2 \hat{i} + \hat{j} + 4 \hat{k}\] are 0, 5 and 8 respectively. Find the vector.

 Dot products of a vector with vectors \[\hat{i} - \hat{j} + \hat{k} , 2\hat{ i} + \hat{j} - 3\hat{k} \text{ and } \text{i} + \hat{j} + \hat{k}\]  are respectively 4, 0 and 2. Find the vector.

If  \[\hat{a} \text{ and } \hat{b}\] are unit vectors inclined at an angle θ, prove that \[\cos\frac{\theta}{2} = \frac{1}{2}\left| \hat{a} + \hat{b} \right|\] 

If \[\hat{a} \text{ and } \hat{b}\] \[\tan\frac{\theta}{2} = \frac{\left| \hat{a} -\hat{b} \right|}{\left| \hat{a} + \hat{b} \right|}\] 

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is \[\sqrt{3}\].

If \[\vec{a,} \vec{b,} \vec{c}\] are three mutually perpendicular unit vectors, then prove that \[\left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{3}\]

If \[\left| \vec{a} + \vec{b} \right| = 60, \left| \vec{a} - \vec{b} \right| = 40 \text{ and } \left| \vec{b} \right| = 46, \text{ find } \left| \vec{a} \right|\]

Show that the vector \[\hat{i} + \hat{j} + \hat{k}\] is equally inclined to the coordinate axes. 

Show that the vectors \[\vec{a} = \frac{1}{7}\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right), \vec{b} = \frac{1}{7}\left( 3 hat{i} - 6 hat{j} + 2 \hat{k} \right), \vec{c} = \frac{1}{7}\left( 6 \hat{i} + 2 \hat{j} - 3 \hat{k} \right)\] mutually perpendicular unit vectors. 

For any two vectors \[\vec{a} \text{ and } \vec{b}\] show that \[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 0 \Leftrightarrow \left| \vec{a} \right| = \left| \vec{b} \right|\]

If \[\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}\]  \[\vec{b} = \hat{i} + \hat{j} - 2 \hat{k}\]  \[\vec{c} = \hat{i} + 3 \hat{j} - \hat{k}\] find λ such that \[\vec{a}\] is perpendicular to \[\lambda \vec{b} + \vec{c}\]  

If \[\vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then find the value of λ, so that \[\vec{p} + \vec{q}\] and \[\vec{p} - \vec{q}\]  are perpendicular vectors. 

If \[\vec{\alpha} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \text{ and } \vec{\beta} = 2 \hat{i} + \hat{j} - 4 \hat{k} ,\] then express \[\vec{\beta}\] in the form of  \[\vec{\beta} = \vec{\beta_1} + \vec{\beta_2} ,\]  where \[\vec{\beta_1}\] is parallel to \[\vec{\alpha} \text{ and } \vec{\beta_2}\]  is perpendicular to \[\vec{\alpha}\]

If either \[\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0}\]  then \[\vec{a} \cdot \vec{b} = 0 .\] But the converse need not be true. Justify your answer with an example. 

Show that the vectors \[\vec{a} = 3 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} - 3 \hat{j} + 5 \hat{k} , \vec{c} = 2 \hat{i} + \hat{j} - 4 \hat{k}\] form a right-angled triangle. 

If \[\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = - \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{c} = 3 \hat{i} + \hat{j}\] \[\vec{a} + \lambda \vec{b}\] is perpendicular to \[\vec{c}\] then find the value of λ. 

Find the angles of a triangle whose vertices are A (0, −1, −2), B (3, 1, 4) and C (5, 7, 1). 

Find the magnitude of two vectors \[\vec{a} \text{ and } \vec{b}\] that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.

Show that the points whose position vectors are \[\vec{a} = 4 \hat{i} - 3 \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - 4 \hat{j} + 5 \hat{k} , \vec{c} = \hat{i} - \hat{j}\] form a right triangle. 

If the vertices A, B and C of ∆ABC have position vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of ∠ABC? 

If A, B and C have position vectors (0, 1, 1), (3, 1, 5) and (0, 3, 3) respectively, show that ∆ ABC is right-angled at C. 

Find the projection of \[\vec{b} + \vec{c}  \text { on }\vec{a}\]  where \[\vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \text{ and } \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .\]

If \[\vec{a} = 5 \hat{i} - \hat{j} - 3 \hat{k} \text{ and } \vec{b} = \hat{i} + 3 \hat{j} - 5 hat{k} ,\] then show that the vectors \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} \] are orthogonal.

A unit vector \[\vec{a}\] makes angles \[\frac{\pi}{4}and\frac{\pi}{3}\] with \[\hat{i}\] and\[\hat{j}\]  respectively and an acute angle θ with \[\hat{k}\] Find the angle θ and components of \[\vec{a}\] 

If two vectors \[\vec{a} \text{ and } \vec{b}\] are such that \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 1 \text{ and } \vec{a} \cdot \vec{b} = 1,\]  then find the value of \[\left( 3 \vec{a} - 5 \vec{b} \right) \cdot \left( 2 \vec{a} + 7 \vec{b} \right) .\] 

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 8\] 

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 12\] 

Find \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 12 \text{ and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]

Find  \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 8 \text{ and } \left| \vec{a} \right| = 8\left| \vec{b} \right|\]

Find \[\left| \vec{a} \right| and \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 3 and \left| \vec{a} \right| = 2\left| \vec{b} \right|\]

Find \[\left| \vec{a} - \vec{b} \right|\] if 

\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} \cdot \vec{b} = 8\]

Find \[\left| \vec{a} - \vec{b} \right|\]  

\[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 4 \text{ and } \vec{a} \cdot \vec{b} = 1\] 

Find \[\left| \vec{a} - \vec{b} \right|\] if  

\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 4\]

Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] if 

\[\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = 2 \text{ and } \vec{a} \cdot \vec{b} = \sqrt{6}\] 

Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\]  

\[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 1\]

Express the vector \[\vec{a} = 5 \text{i} - 2 \text{j} + 5 \text{k}\] as the sum of two vectors such that one is parallel to the vector \[\vec{b} = 3 \text{i} + \text{k}\]  and other is perpendicular to \[\vec{b}\]

If \[\vec{a} \text{ and } \vec{b}\] are two vectors of the same magnitude inclined at an angle of 30°, such that \[\vec{a} \cdot \vec{b} = 3, \text{ find } \left| \vec{a} \right|, \left| \vec{b} \right| .\] 

Express \[2 \hat{i} - \hat{j} + 3 \hat{k}\] as the sum of a vector parallel and a vector perpendicular to \[2 \hat{i} + 4 \hat{j} - 2 \hat{k} .\] 

Decompose the vector \[6 \hat{i} - 3 \hat{j} - 6 \hat{k}\] into vectors which are parallel and perpendicular to the vector \[\hat{i} + \hat{j} + \hat{k} .\] 

Let \[\vec{a} = 5 \hat{i} - \hat{j} + 7 \hat{k} \text{ and } \vec{b} = \hat{i} - \hat{j} + \lambda \hat{k} .\] Find λ such that \[\vec{a} + \vec{b}\] is orthogonal to \[\vec{a} - \vec{b}\] 

If \[\vec{a} \cdot \vec{a} = 0 \text{ and } \vec{a} \cdot \vec{b} = 0,\] what can you conclude about the vector \[\vec{b}\] ?

If \[\vec{c}\] s perpendicular to both \[\vec{a} \text{ and } \vec{b}\] then prove that it is perpendicular to both \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b}\] 

If \[\left| \vec{a} \right| = a \text{ and } \left| \vec{b} \right| = b,\] prove that \[\left( \frac{\vec{a}}{a^2} - \frac{\vec{b}}{b^2} \right)^2 = \left( \frac{\vec{a} - \vec{b}}{ab} \right)^2 .\] 

If \[\vec{a,} \vec{b,} \vec{c}\]  are three non-coplanar vectors, such that \[\vec{d} \cdot \vec{a} = \vec{d} \cdot \vec{b} = \vec{d} \cdot \vec{c} = 0,\] then show that \[\vec{d}\] is the null vector.

If a vector \[\vec{a}\] is perpendicular to two non-collinear vectors \[\vec{b} \text{ and } \vec{c} , \text{ then show that } \vec{a}\] is perpendicular to every vector in the plane of \[\vec{b} \text{ and } \vec{c} .\] 

If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} ,\] show that the angle θ between the vectors \[\vec{b} \text{ and } \vec{c}\] is given by  \[\frac{\left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 - \left| \vec{c} \right|^2}{2\left| \vec{b} \right| \left| \vec{c} \right|} .\]

Let \[\vec{u,} \vec{v} \text{ and } \vec{w}\]  be vectors such \[\vec{u} + \vec{v} + \vec{w} = \vec{0} .\] If \[\left| \vec{u} \right| = 3, \left| \vec{v} \right| = 4 \text{ and } \left| \vec{w} \right| = 5,\] then find \[\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} .\]

Let \[\vec{a} = x^2 \hat{i} + 2 \hat{j} - 2 \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{c} = x^2 \hat{i} + 5 \hat{j} - 4 \hat{k}\] be three vectors. Find the values of x for which the angle between \[\vec{a} \text{ and } \vec{b}\ \text{ is acute and the angle between \[\vec{b} \text{ and } \vec{c}\] is obtuse.

Find the values of x and y if the vectors \[\vec{a} = 3 \hat{i} + x \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{j} + y \hat{k}\] are mutually perpendicular vectors of equal magnitude. 

If \[\vec{a}\] \[\vec{b}\]  are two vectors such that \[\left| \vec{a} + \vec{b} \right| = \left| \vec{b} \right|\] then prove that \[\vec{a} + 2 \vec{b}\] is perpendicular to \[\vec{a}\] 

In a triangle OAB,\[\angle\]AOB = 90º. If P and Q are points of trisection of AB, prove that \[{OP}^2 + {OQ}^2 = \frac{5}{9} {AB}^2\]

Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus. 

(Pythagoras's Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 

Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus. 

Prove that the diagonals of a rhombus are perpendicular bisectors of each other. 

Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square. 

If AD is the median of ∆ABC, using vectors, prove that \[{AB}^2 + {AC}^2 = 2\left( {AD}^2 + {CD}^2 \right)\] 

If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles. 

In a quadrilateral ABCD, prove that \[{AB}^2 + {BC}^2 + {CD}^2 + {DA}^2 = {AC}^2 + {BD}^2 + 4 {PQ}^2\] where P and Q are middle points of diagonals AC and BD. 

What is the angle between vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes 2 and \[\sqrt{3}\] respectively? Given \[\vec{a} . \vec{b} = \sqrt{3} .\]

\[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\vec{a} . \vec{b} = 6, \left| \vec{a} \right| = 3 \text{ and } \left| \vec{b} \right| = 4 .\] Write the projection of \[\vec{a} \text{ on } \vec{b}\] 

If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} , \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 7,\] then the angle between \[\vec{a} \text{ and } \vec{b}\] is 

(a) \[\frac{\pi}{6}\]  

(b) \[\frac{2\pi}{3}\] 

(c) \[\frac{5\pi}{3}\]

(d) \[\frac{\pi}{3}\] 

Find the cosine of the angle between the vectors \[4 \hat{i} - 3 \hat{j} + 3 \hat{k} \text{ and } 2 \hat{i} - \hat{j} - \hat{k}] .\] 

If the vectors \[3 \hat{i} + m \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} - 8 \hat{k}\]  are orthogonal, find m. 

If the vectors \[3 \hat{i} - 2 \hat{j} - 4 \hat{k}\text{ and } 18 \hat{i} - 12 \hat{j}] - m \hat{k}\] are parallel, find the value of m.

If \[\vec{a} \text{ and } \vec{b}\] are vectors of equal magnitude, write the value of \[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) .\] 

If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0,\] find the relation between the magnitudes of \[\vec{a} \text{ and } \vec{b}\]  

For any two vectors \[\vec{a} \text{ and } \vec{b}\] write when \[\left| \vec{a} + \vec{b} \right| = \left| \vec{a} \right| + \left| \vec{b} \right|\] holds. 

For any two vectors \[\vec{a} \text{ and } \vec{b}\] write when \[\left| \vec{a} + \vec{b} \right| = \left| \vec{a} - \vec{b} \right|\] holds. 

If \[\vec{a} \text{ and } \vec{b}\] are two vectors of the same magnitude inclined at an angle of 60° such that \[\vec{a} . \vec{b} = 8,\] write the value of their magnitude. 

If \[\vec{a} . \vec{a} = 0 \text{ and } \vec{a} . \vec{b} = 0,\] what can you conclude about the vector \[\vec{b}\] 

If \[\vec{b}\] is a unit vector such that\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 8, \text{ find } \left| \vec{a} \right| .\]

If \[\hat{a} , \hat{b}\] are unit vectors such that \[\hat{a} + \hat{b}\]  is a unit vector, write the value of \[\left| \hat{a} - \hat{b} \right| .\] 

If \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2, \text{ find } \left| \vec{a} - \vec{b} \right| .\]

If \[\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + \hat{k} ,\]  find the projection of \[\vec{a} \text{ on } \vec{b}\] 

For any two non-zero vectors, write the value of \[\frac{\left| \vec{a} + \vec{b} \right|^2 + \left| \vec{a} - \vec{b} \right|^2}{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2} .\] 

Write the projections of \[\vec{r} = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}\] on the coordinate axes. 

Write the component of \[\vec{b}\] along \[\vec{a}\] 

Write the value of \[\left( \vec{a} . \hat{i} \right) \hat{i} + \left( \vec{a} . \hat{j} \right) \hat{j} + \left( \vec{a} . \hat{k} \right) \hat{k} ,\]  where \[\vec{a}\] is any vector. 

Find the value of θ ∈(0, π/2) for which vectors \[\vec{a} = \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \text{ and } \vec{b} = \hat{i} - \sqrt{3} \hat{j} + 2 \hat{k}\] are perpendicular.

Write the projection of \[\hat{i} + \hat{j} + \hat{k}\] along the vector \[\hat{j}\] 

Write a vector satisfying \[\vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1 .\]

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, find the angle between \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} .\]

If \[\vec{a} \text{ and } \vec{b}\] are mutually perpendicular unit vectors, write the value of \[\left| \vec{a} + \vec{b} \right| .\] 

If \[\vec{a} , \vec{b} \text{ and } \vec{c}\] are mutually perpendicular unit vectors, write the value of \[\left| \vec{a} + \vec{b} + \vec{c} \right| .\] 

Find the angle between the vectors \[\vec{a} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - \hat{k} .\]

For what value of λ are the vectors \[\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] perpendicular to each other?

Find the projection of \[\vec{a} \text{ on } \vec{b} \text{ if } \vec{a} \cdot \vec{b} = 8 \text{ and } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} .\] 

Write the value of p for which \[\vec{a} = 3 \hat{i} + 2 \hat{j}] + 9 \hat{k} \text{ and } \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k}\]

Find the value of λ if the vectors \[2 \hat{i} + \lambda \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 2 \hat{j} - 4 \hat{k}\] are perpendicular to each other. 

If \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 3,\] find the projection of \[\vec{b} \text{ on } \vec{a}\] 

Write the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes \[\sqrt{3}\] and 2 respectively if \[\vec{a} \cdot \vec{b} = \sqrt{6} .\]

Write the projection of the vector \[\hat{i} + 3 \hat{j} + 7 \hat{k}\] on the vector \[2 \hat{i} - 3 \hat{j} + 6 \hat{k}\] 

Find λ when the projection of \[\vec{a} = \lambda \hat{i} + \hat{j} + 4 \hat{k} \text{ on } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k}\]  is 4 units. 

For what value of λ are the vectors \[\vec{a} = 2 \text{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] perpendicular to each other?

Write the projection of the vector \[7 \hat{i} + \hat{j} - 4 \hat{k}\] on the vector \[2 \hat{i} + 6 \hat{j}+ 3 \hat{k} .\] 

Write the value of λ so that the vectors \[\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] are perpendicular to each other. 

Write the projection of \[\vec{b} + \vec{c} \text{ on } \vec{a} \text{ when } \vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \text{ and } \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .\] 

If \[\vec{a}\] and \[\vec{b}\] are perpendicular vectors, \[\left| \vec{a} + \vec{b} \right| = 3\] and \[\left| \vec{a} \right| = 5\] find the value of \[\left| \vec{b} \right|\]

If the vectors \[\vec{a}\]  and \[\vec{b}\] are such that \[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = \frac{2}{3}\] and \[\vec{a} \times \vec{b}\] is a unit vector, then write the angle between \[\vec{a}\] and \[\vec{b}\] 

If \[\vec{a}\] and \[\vec{b}\] are two unit vectors such that \[\vec{a} + \vec{b}\] is also a unit vector, then find the angle between \[\vec{a}\] and \[\vec{b}\] 

If \[\vec{a}\] and \[\vec{b}\] are unit vectors, then find the angle between \[\vec{a}\] and \[\vec{b}\] given that \[\left( \sqrt{3} \vec{a} - \vec{b} \right)\] is a unit vector.      

The vectors \[\vec{a} \text{ and } \vec{b}\] satisfy the equations \[2 \vec{a} + \vec{b} = \vec{p} \text{ and } \vec{a} + 2 \vec{b} = \vec{q} , \text{ where } \vec{p} = \hat{i} + \hat{j} \text{ and } \vec{q} = \hat{i} - \hat{j} .\] the angle between \[\vec{a} \text{ and } \vec{b}\] then 

(a) \[\cos \theta = \frac{4}{5}\]

(b) \[\sin \theta = \frac{1}{\sqrt{2}}\]

(c) \[\cos \theta = - \frac{4}{5}\]

(d) \[\cos \theta = - \frac{3}{5}\] 

If \[\vec{a} \cdot \text{i} = \vec{a} \cdot \left( \hat{i} + \hat{j} \right) = \vec{a} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 1,\]  then \[\vec{a} =\] 

(a) \[\vec{0}\] 

(b) \[\hat{i}\]  

(c)  \[\hat{j}\]

(d) \[\hat{i} + \hat{j} + \hat{k}\] 

If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} , \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 7,\] then the angle between \[\vec{a} \text{ and } \vec{b}\] is 

(a) \[\frac{\pi}{6}\] 

(b) \[\frac{2\pi}{3}\] 

(c) \[\frac{5\pi}{3}\] 

(d) \[\frac{\pi}{3}\]  

Let \[\vec{a} \text{ and } \vec{b}\]  be two unit vectors and α be the angle between them. Then, \[\vec{a} + \vec{b}\] is a unit vector if 

(a) \[\vec{a} + \vec{b}\] 

(b) \[\alpha = \frac{\pi}{3}\] 

(c) \[\alpha = \frac{2\pi}{3}\] 

(d) \[\alpha = \frac{\pi}{2}\]

The vector (cos α cos β) \[\hat{i}\] + (cos α sin β) \[\hat{j}\] + (sin α) \[\hat{k}\]  is a 

(a) null vector 

(b) unit vector 

(c) constant vector 

(d) None of these 

If the position vectors of P and Q are \[\hat{i} + 3 \hat{j} - 7 \hat{k} \text{ and } 5 \text{i} - 2 \hat{j} + 4 \hat{k}\] then the cosine of the angle between \[\vec{PQ}\] and y-axis is 

(a) \[\frac{5}{\sqrt{162}}\] 

(b) \[\frac{4}{\sqrt{162}}\] 

(c) \[- \frac{5}{\sqrt{162}}\] 

(d) \[\frac{11}{\sqrt{162}}\] 

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then which of the following values of \[\vec{a} . \vec{b}\] is not possible? 

(a) \[\sqrt{3}\] 

(b) \[\sqrt{3}/2\] 

(c) \[1/\sqrt{2}\] 

(d) −1/2 

If the vectors \[\hat{i} - 2x \hat{j} + 3y \hat{k} \text{ and } \hat{i} + 2x \hat{j} - 3y \hat{k}\] are perpendicular, then the locus of (x, y) is

(a) a circle 

(b) an ellipse 

(c) a hyperbola 

(d) None of these 

The vector component of \[\vec{b}\] perpendicular to \[\vec{a}\] is 

(a) \[\left( \vec{b} . \vec{c} \right) \vec{a}\] 

(b) \[\frac{\vec{a} \times \left( \vec{b} \times \vec{a} \right)}{\left| \vec{a} \right|^2}\] 

(c) \[\vec{a} \times \left( \vec{b} \times \vec{a} \right)\] 

(d) None of these 

What is the length of the longer diagonal of the parallelogram constructed on \[5 \vec{a} + 2 \vec{b} \text{ and } \vec{a} - 3 \vec{b}\] if it is given that \[\left| \vec{a} \right| = 2\sqrt{2}, \left| \vec{b} \right| = 3\] and the angle between \[\vec{a} \text{ and } \vec{b}\] is π/4? 

(a) 15 

(b) \[\sqrt{113}\] 

(c) \[\sqrt{593}\] 

(d) \[\sqrt{369}\] 

If \[\vec{a}\] is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ \[\vec{a}\] is a unit vector if 

(a) λ = 1 

(b) λ = −1 

(c) a = |λ| 

(d) \[a = \frac{1}{\left| \lambda \right|}\] 

If θ is the angle between two vectors \[\vec{a} \text{ and } \vec{b} , \text{ then } \vec{a} \cdot \vec{b} \geq 0\]  only when 

(a) \[0 < \theta < \frac{\pi}{2}\] 

(b) \[0 \leq \theta \leq \frac{\pi}{2}\] 

(c) 0 < θ < π 

(d) 0 ≤ θ ≤ π 

The values of x for which the angle between \[\vec{a} = 2 x^2 \hat{i} + 4x \hat{j} + \hat{k} , \vec{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}\]  is obtuse and the angle between \[\vec{b}\] and the z-axis is acute and less than \[\frac{\pi}{6}\] 

(a) \[x > \frac{1}{2} or x < 0\]

(b) \[0 < x < \frac{1}{2}\] 

(c) \[\frac{1}{2} < x < 15\] 

(d) ϕ 

If \[\vec{a} , \vec{b} , \vec{c}\] are any three mutually perpendicular vectors of equal magnitude a, then \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] is equal to 

(a) a 

(b) \[\sqrt{2}a\] 

(c) \[\sqrt{3}a\] 

(d) 2a 

(e) None of these 

If the vectors \[3 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} + 8 \hat{k}\] are perpendicular, then λ is equal to 

(a) −14 

(b) 7 

(c) 14 

(d) \[\frac{1}{7}\] 

The projection of the vector \[\hat{i} + \hat{j} + \hat{k}\] along the vector of \[\hat{j}\] 

(a) 1 

(b) 0 

(c) 2 

(d) −1 

(e) −2 

The vectors \[2 \hat{i} + 3 \hat{j} - 4 \hat{k}\] and \[a \hat{i} + \hat{b} j + c \hat{k}\] are perpendicular if 

(a) a = 2, b = 3, c = −4 

(b) a = 4, b = 4, c = 5 

(c) a = 4, b = 4, c = −5 

(d) a = −4, b = 4, c = −5 

If \[\left| \vec{a} \right| = \left| \vec{b} \right|, \text{ then } \left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) =\]

(a) positive 

(b) negative 

(c) 0 

(d) None of these 

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors inclined at an angle θ, then the value of \[\left| \vec{a} - \vec{b} \right|\] 

(a) \[2 \sin\frac{\theta}{2}\] 

(b) 2 sin θ 

(c) \[2 \cos\frac{\theta}{2}\] 

(d) 2 cos θ 

If \[\vec{a} and \vec{b}\] are unit vectors, then the greatest value of \[\sqrt{3}\left| \vec{a} + \vec{b} \right| + \left| \vec{a} - \vec{b} \right|\] 

(a) 2 

(b) \[2\sqrt{2}\] 

(c) 4 

(d) None of these 

If the angle between the vectors \[x \hat{i} + 3 \hat{j}- 7 \hat{k} \text{ and } x \hat{i} - x \hat{j} + 4 \hat{k}\] is acute, then x lies in the interval 

(a) (−4, 7) 

(b) [−4, 7] 

(c) R −[−4, 7] 

(d) R −(4, 7) 

If \[\vec{a} \text{ and } \vec{b}\] are two unit vectors inclined at an angle θ, such that \[\left| \vec{a} + \vec{b} \right| < 1,\] then 

(a) \[\theta < \frac{\pi}{3}\] 

(b) \[\theta > \frac{2\pi}{3}\] 

(c) \[\frac{\pi}{3} < \theta < \frac{2\pi}{3}\] 

(d) \[\frac{2\pi}{3} < \theta < \pi\] 

Let \[\vec{a} , \vec{b} , \vec{c}\] be three unit vectors, such that \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] =1 and \[\vec{a}\] is perpendicular to \[\vec{b}\]  If \[\vec{c}\] makes angles α and β with \[\vec{a} and \vec{b}\] respectively, then cos α + cos β =

(a) \[- \frac{3}{2}\]

(b) \[\frac{3}{2}\]

(c) 1 

(d) −1

The orthogonal projection of \[\vec{a} \text{ on } \vec{b}\] is 

(a) \[\frac{\left( \vec{a} \cdot \vec{b} \right) \vec{a}}{\left| \vec{a} \right|^2}\] 

(b) \[\frac{\left( \vec{a} \cdot \vec{b} \right) \vec{b}}{\left| \vec{b} \right|^2}\] 

(c)  \[\frac{\vec{a}}{\left| \vec{a} \right|}\] 

(d) \[\frac{\vec{b}}{\left| \vec{b} \right|}\] 

If θ is an acute angle and the vector (sin θ) \[\text{i}\]  + (cos θ) \[\hat{j}\]  is perpendicular to the vector \[\hat{i} - \sqrt{3} \hat{j} ,\] then θ = 

(a) \[\frac{\pi}{6}\] 

(b) \[\frac{\pi}{5}\] 

(c)  \[\frac{\pi}{4}\] 

(d)  \[\frac{\pi}{3}\]

If \[\vec{a} \text{ and }\vec{b}\] be two unit vectors and θ the angle between them, then \[\vec{a} + \vec{b}\] is a unit vector if θ = 

(a) \[\frac{\pi}{4}\] 

(b) \[\frac{\pi}{3}\] 

(c) \[\frac{\pi}{2}\] 

(d) \[\frac{2\pi}{3}\]