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RD Sharma solutions for Class 12 Mathematics chapter 25 - Vector or Cross Product

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) - Shaalaa.com

Chapter 25: Vector or Cross Product

Ex. 25.1very short answersMCQ

Chapter 25: Vector or Cross Product Exercise 25.1 solutions [Pages 29 - 31]

Ex. 25.1 | Q 1 | Page 29

\[\text{ If } \vec{a} = \hat { i }  + 3 \hat { j }  - 2 \hat { k } \text{ and }  \vec{b} = - \hat { i }  + 3 \hat { k }   , \text{ find }  \left| \vec{a} \times \vec{b} \right| .\]

Ex. 25.1 | Q 2.1 | Page 29

If \[\vec{a} = 3 \hat { i } + 4 \hat { j } \text{ and }  \vec{b} = \hat { i  } + \hat{ j }  + \hat{ k } ,\]  find the value of \[\left| \vec{a} \times \vec{b} \right| .\]

 
Ex. 25.1 | Q 2.2 | Page 29

If \[\vec{a} = 2 \hat{ i } + \hat{ k }  , \vec{b} = \hat { i }  + \hat{ j } + \hat{ k }  ,\]  find the magnitude of  \[\vec{a} \times \vec{b} .\]

 

 

Ex. 25.1 | Q 3.1 | Page 29

 Find a unit vector perpendicular to both the vectors  \[4 \hat{ i } - \hat{ j }  + 3 \hat{ k } \text{ and }  - 2 \hat{ i  } + \hat{ j }  - 2 \hat{ k }  .\]

 

Ex. 25.1 | Q 3.2 | Page 29

Find a unit vector perpendicular to the plane containing the vectors  \[\vec{a} = 2 \hat{ i } + \hat{ j }  + \hat{ k } \text{ and }  \vec{b} = \hat{ i } + 2 \hat{ j }  + \hat{ k } .\]

 

Ex. 25.1 | Q 4 | Page 29

Find the magnitude of \[\vec{a} = \left( 3 \hat{ k }  + 4 \hat{ j } \right) \times \left( \hat{ i }  + \hat{ j }  - \hat{ k }  \right) .\]

 
Ex. 25.1 | Q 5 | Page 29
\[\text{ If }  \vec{a} = 4 \hat{ i }  + 3 \hat{ j }  + \hat{ k }  \text{ and }  \vec{b} = \hat{ i }  - 2 \hat{ k } ,\text{  then find }  \left| 2 \hat{ b } \times \vec{a} \right| .\]

 

Ex. 25.1 | Q 6 | Page 29
\[\text{ If }  \vec{ a } = 3 \hat{ i }- \hat{ j }  - 2 \hat{ k } \text{  and } \vec{b} = 2 \hat{ i }  + 3 \hat{ j } + \hat{ k }  , \text{ find }  \left( \vec{a} + 2 \vec{b} \right) \times \left( 2 \vec{a} - \vec{b} \right) .\]

 

Ex. 25.1 | Q 7.1 | Page 29

Find a vector of magnitude 49, which is perpendicular to both the vectors  \[2 \hat{ i }   + 3 \hat{ j }  + 6 \hat{ k }  \text{ and } 3 \hat{ i }  - 6 \hat{ j }  + 2 \hat{ k }  .\]

 

Ex. 25.1 | Q 7.2 | Page 29

Find a vector whose length is 3 and which is perpendicular to the vector \[\vec{a} = 3 \hat{ i }  + \hat{ j  } - 4 \hat{ k }  \text{ and }  \vec{b} = 6 \hat{ i }  + 5 \hat{ j }  - 2 \hat{ k } .\]

Ex. 25.1 | Q 8.1 | Page 29

Find the area of the parallelogram determined by the vector \[2 \hat{ i }  \text{ and }  3 \hat{ j } \] .

 

Ex. 25.1 | Q 8.2 | Page 29

Find the area of the parallelogram determined by the vector \[2 \hat{ i } + \hat{ j } + 3 \hat{ k }  \text{ and }  \hat{ i }  - \hat{ j } \] .

 

Ex. 25.1 | Q 8.3 | Page 29

Find the area of the parallelogram determined by the vector \[3 \hat{ i } + \hat{ j }  - 2 \hat{ k } \text{  and }  \hat{ i }  - 3 \hat{ j }  + 4 \hat{ k } \] .

 

Ex. 25.1 | Q 8.4 | Page 29

Find the area of the parallelogram determined by the vector \[\hat{ i }  - 3 \hat{ j } + \hat{ k }  \text{ and }  \hat{ i }  + \hat{ j } + \hat{ k }  .\]

 

Ex. 25.1 | Q 9.1 | Page 30

Find the area of the parallelogram whose diagonals are  \[4 \hat{ i } - \hat{ j }  - 3 \hat{ k }  \text{ and }  - 2 \hat{ j }  + \hat{ j }  - 2 \hat{ k } \]

 

Ex. 25.1 | Q 9.2 | Page 30

Find the area of the parallelogram whose diagonals are  \[2 \hat{ i }+ \hat{ k } \text{ and } \hat{ i } + \hat{ j } + \hat{ k } \]

 

Ex. 25.1 | Q 9.3 | Page 30

Find the area of the parallelogram whose diagonals are  \[3 \hat{ i }  + 4 \hat{ j }  \text{ and } \hat{ i } + \hat{ j } + \hat{ k }\]

 

Ex. 25.1 | Q 9.4 | Page 30

Find the area of the parallelogram whose diagonals are \[2 \hat{ i }  + 3 \hat{ j } + 6 \hat{ k } \text{ and }  3 \hat{ i }  - 6 \hat{ j }  + 2 \hat{ k } \]

 

Ex. 25.1 | Q 10 | Page 30

If \[\vec{a} = 2 \hat{ i }  + 5 \hat{ j }  - 7 \hat{ k }  , \vec{b} = - 3 \hat{ i } + 4 \hat{ j }  + \hat{ k }  \text{ and } \vec{c} = \hat{ i }  - 2 \hat{ j }  - 3 \hat{ k }  ,\] compute \[\left( \vec{a} \times \vec{b} \right) \times \vec{c} \text{ and }  \vec{a} \times \left( \vec{b} \times \vec{c} \right)\]  and verify that these are not equal.

 
 
 
Ex. 25.1 | Q 11 | Page 30
\[\text{ If }  \left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and }  \left| \vec{a} \times \vec{b} \right| = 8, \text { find }  \vec{a} \cdot \vec{b} .\]

 

Ex. 25.1 | Q 12 | Page 30

Given \[\vec{a} = \frac{1}{7}\left( 2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k }  \right), \vec{b} = \frac{1}{7}\left( 3 \hat{ i } - 6 \hat{ j }  + 2 \hat{ k }  \right), \vec{c} = \frac{1}{7}\left( 6 \hat{ i } + 2 \hat{ j }  - 3 \hat{ k }\right), \hat{ i } , \hat{ j }  , \hat{ k } \] being a right handed orthogonal system of unit vectors in space, show that \[\vec{a} , \vec{b} , \vec{c}\] is also another system.

 
 
Ex. 25.1 | Q 13 | Page 30
\[\text{ If }  \left| \vec{a} \right| = 13, \left| \vec{b} \right| = 5 \text{ and }  \vec{a} . \vec{b} = 60, \text{ then find }  \left| \vec{a} \times \vec{b} \right| .\]

 

Ex. 25.1 | Q 14 | Page 30

Find the angle between two vectors \[\vec{a} \text{ and }  \vec{b}\] , if \[\left| \vec{a} \times \vec{b} \right| = \vec{a} \cdot \vec{b} .\]

 
Ex. 25.1 | Q 15 | Page 30

if \[\vec{a} \times \vec{b} = \vec{b} \times \vec{c} \neq 0,\]  then  show that \[\vec{a} + \vec{c} = m \vec{b} ,\]  where m is any scalar.

 
 

 

Ex. 25.1 | Q 16 | Page 30

if \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 7 \text{ and }  \vec{a} \times \vec{b} = 3 \hat{ i }  + 2 \hat{ j } + 6 \hat{ k } ,\]  find the angle between  \[\vec{a} \text{ and }  \vec{b} .\]

 

Ex. 25.1 | Q 17 | Page 30

What inference can you draw if \[\vec{a} \times \vec{b} = \vec{0} \text{ and }  \vec{a} \cdot \vec{b} = 0 .\]

 
Ex. 25.1 | Q 18 | Page 30

If \[\vec{a,} \vec{b,} \vec{c}\] are three unit vectors such that \[\vec{a} \times \vec{b} = \vec{c} , \vec{b} \times \vec{c} = \vec{a,} \vec{c} \times \vec{a} = \vec{b} .\]  Show that \[\vec{a,} \vec{b,} \vec{c}\] form an orthonormal right handed triad of unit vectors.

 
 
 

 

Ex. 25.1 | Q 19 | Page 30

Find a unit vector perpendicular to the plane ABC, where the coordinates of AB and Care A (3, −1, 2), B (1, −1, −3) and C (4, −3, 1).

Ex. 25.1 | Q 20 | Page 30

If abc are the lengths of sides, BCCA and AB of a triangle ABC, prove that \[\vec{BC} + \vec{CA} + \vec{AB} = \vec{0}\]  and deduce that \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]

 
 
Ex. 25.1 | Q 21 | Page 30

if \[\vec{a} = \hat{ i }- 2\hat{ j }  + 3 \hat{ k }  , \text{ and }  \vec{b} = 2 \hat{ i }  + 3 \hat{ j }  - 5 \hat{ k }  ,\]  then find \[\vec{a} \times \vec{b} .\]  Verify th at \[\vec{a} \text{ and }  \vec{a} \times \vec{b}\] are perpendicular to each other.

 
 
 

 

Ex. 25.1 | Q 22 | Page 30

If  \[\vec{p} \text{ and }  \vec{q}\] are unit vectors forming an angle of 30°; find the area of the parallelogram having \[\vec{a} = \vec{p} + 2 \vec{q} \text{ and }  \vec{b} = 2 \vec{p} + \vec{q}\] as its diagonals.

 
 

 

Ex. 25.1 | Q 23 | Page 30

For any two vectors \[\vec{a} \text{ and }  \vec{b}\] , prove that \[\left| \vec{a} \times \vec{b} \right|^2 = \begin{vmatrix}\vec{a} . \vec{a} & & \vec{a} . \vec{b} \\ \vec{b} . \vec{a} & & \vec{b} . \vec{b}\end{vmatrix}\]

 
 
Ex. 25.1 | Q 24 | Page 30

Define  \[\vec{a} \times \vec{b}\] and prove that \[\left| \vec{a} \times \vec{b} \right| = \left( \vec{a} . \vec{b} \right)\] tan θ, where θ is the angle between \[\vec{a} \text{ and }  \vec{b}\] .

 
 

 

Ex. 25.1 | Q 25 | Page 30
\[\text{ If }  \left| \vec{a} \right| = \sqrt{26}, \left| \vec{b} \right| = 7 \text{ and }  \left| \vec{a} \times \vec{b} \right| = 35, \text{ find }  \vec{a} . \vec{b} .\]

 

Ex. 25.1 | Q 26 | Page 30

Find the area of the triangle formed by OAB when \[\vec{OA} = \hat{ i } + 2 \hat{ j }  + 3 \hat{ k }  , \vec{OB} = - 3 \hat{ i }  - 2 \hat{ j }+ \hat{ k }  .\]

Ex. 25.1 | Q 27 | Page 30

Let \[\vec{a} = \hat{ i } + 4 \hat{ j }  + 2 \hat{ k } , \vec{b} = 3 \hat{ i }- 2 \hat{ j } + 7 \hat{ k }  \text{ and } \vec{c} = 2 \hat{ i } - \hat{ j }  + 4 \hat{ k }  .\]  Find a vector \[\vec{d}\] which is perpendicular to both \[\vec{a} \text{ and } \vec{d}\] \[\text{ and }  \vec{c} \cdot \vec{d} = 15 .\]

 
 

 

Ex. 25.1 | Q 28 | Page 31

Find a unit vector perpendicular to each of the vectors \[\vec{a} + \vec{b} \text{ and }  \vec{a} - \vec{b} , \text{ where }  \vec{a} = 3 \hat{ i }  + 2 \hat{ j }  + 2 \hat{ k }  \text{ and }  \vec{b} = \hat{ i } + 2 \hat{ j }  - 2 \hat{ k }  .\]

 
Ex. 25.1 | Q 29 | Page 31

Using vectors find the area of the triangle with vertices, A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).

Ex. 25.1 | Q 30 | Page 31

If \[\vec{a} = 2 \hat{ i } - 3 \hat{ j  } + \hat{ k } , \vec{b} = -\hat{  i }  + \hat{ k } , \vec{c} = 2 \hat{ j }  - \hat{ k } \]  are three vectors, find the area of the parallelogram having diagonals \[\left( \vec{a} + \vec{b} \right)\]  and \[\left( \vec{b} + \vec{c} \right)\] .

 
 
Ex. 25.1 | Q 31 | Page 31

The two adjacent sides of a parallelogram are \[2 \hat{ i  } - 4 \hat{ j }  + 5 \hat{ k }  \text{ and }  \hat{ i } - 2 \hat{ j }  - 3\hat{ k }  .\]\  Find the unit vector parallel to one of its diagonals. Also, find its area. 

 
 
Ex. 25.1 | Q 32 | Page 31

If either  \[\vec{a} = \vec{0} \text{ or }  \vec{b} = \vec{0} , \text{ then }  \vec{a} \times \vec{b} = \vec{0} .\]  Is the converse true? Justify your answer with an example.

 
Ex. 25.1 | Q 33 | Page 31

If \[\vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k }  , \vec{b} = b_1 \hat{ i }  + b_2 \hat{ j }  + b_3 \hat{ k }  \text{ and }  \vec{c} = c_1 \hat{ i } + c_2 \hat{ j }  + c_3 \hat{ k }  ,\]then verify that \[\vec{a} \times \left(  \vec{b} + \vec{c} \right) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} .\]

Ex. 25.1 | Q 34.1 | Page 31

Using vectors, find the area of the triangle with vertice A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) .

Ex. 25.1 | Q 34.2 | Page 31

Using vectors, find the area of the triangle with vertice A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1)  .    

Ex. 25.1 | Q 35 | Page 31

Find all vectors of magnitude \[10\sqrt{3}\] that are perpendicular to the plane of \[\hat{ i }  + 2 \hat{ j }  + \hat{ k } \] and \[- \hat { i }  + 3 \hat{ j }  + 4 \hat{ k } \] .

 
Ex. 25.1 | Q 36 | Page 31

The two adjacent sides of a parallelogram are `2hati-4hatj-5hatk and 2 hati+2hatj+3hatj` . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

Ex. 25.1 | Q 37 | Page 31

If  \[\left| \vec{a} \times \vec{b} \right|^2 + \left| \vec{a} \cdot \vec{b} \right|^2 = 400\] and  \[\left| \vec{a} \right| = 5,\]  then write the value of \[\left| \vec{b} \right| .\]

 

Chapter 25: Vector or Cross Product Exercise very short answers solutions [Pages 33 - 34]

very short answers | Q 1 | Page 33

Define vector product of two vectors.

 
very short answers | Q 2 | Page 33

Write the value  \[\left( \hat{ i }  \times \hat{ j }  \right) \cdot \hat{ k }  + \hat{ i }  \cdot \hat{ j }  .\]

 

very short answers | Q 3 | Page 33

Write the value of  \[\hat{ i } . \left( \hat{ j } \times \hat{ k }  \right) + \hat{ j }  . \left( \hat{ k } \times \hat{ i }  \right) + \hat{ k }  . \left( \hat{ j }  \times \hat{ i }  \right) .\]

 

very short answers | Q 4 | Page 33

Write the value of  \[\hat{ i } . \left( \hat{ j } \times \hat{ k }  \right) + \hat{ j }  . \left( \hat{ k } \times \hat{ i }  \right) + \hat{ k }  . \left( \hat{ i }  \times \hat{ j }  \right) .\]

 

very short answers | Q 5 | Page 33

Write the value of \[\hat{ i }  × \left( \hat{ j } + \hat{ k }  \right) + \hat{ j }   ×  \left( \hat{ k } + \hat{ i }  \right) + \hat{ k }  ×   \left( \hat{ i }  + \hat{ j }  \right) .\]

 
very short answers | Q 6 | Page 33

Write the expression for the area of the parallelogram having \[\vec{a} \text{ and } \vec{b}\] as its diagonals.

 
 
very short answers | Q 7 | Page 33

For any two vectors  \[\vec{a} \text{ and }  \vec{b}\] write the value of \[\left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2\] in terms of their magnitudes.

 
 
very short answers | Q 8 | Page 33

If \[\vec{a} \text{ and }  \vec{b}\] are two vectors of magnitudes 3 and \[\frac{\sqrt{2}}{3}\]  espectively such that \[\vec{a} \times \vec{b}\] is a unit vector. Write the angle between \[\vec{a} \text{ and }  \vec{b} .\]

 
 
 

 

very short answers | Q 9 | Page 33
\[\text{ If }  \left| \vec{a} \right| = 10, \left| \vec{b} \right| = 2 \text{ and }  \left| \vec{a} \times \vec{b} \right| = 16, \text{ find }  \vec{a} . \vec{b} .\]

 

very short answers | Q 10 | Page 33

For any two vectors \[\vec{a}\] and \[\vec{b}\] , find \[\vec{a} . \left( \vec{b} \times \vec{a} \right) .\]

 
 
 
 
very short answers | Q 11 | Page 33

If   \[\vec{a} \text{ and }  \vec{b}\] are two vectors such that \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3}\text{ and }  \vec{a} . \vec{b} = 1,\]  find the angle between.

 
 

 

very short answers | Q 12 | Page 33

For any three vectors \[\vec{a,} \vec{b} \text{ and }  \vec{c}\] write the value of \[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right) .\]

 
 
very short answers | Q 13 | Page 33

For any two vectors \[\vec{a} \text{ and } \vec{b} , \text{ find } \left( \vec{a} \times \vec{b} \right) . \vec{b} .\]

 
very short answers | Q 14 | Page 33

Write the value of \[\hat{ i }  \times \left(\hat{  j }  \times \hat{ k }  \right) .\]

 
very short answers | Q 15 | Page 33

If \[\vec{a} = 3 \hat{ i }  - \hat{ j }  + 2 \hat{ k } \] and  \[\vec{b} = 2 \hat { i }  + \hat{ j }  - \hat{ k} ,\]  then find \[\left( \vec{a} \times \vec{b} \right) \vec{a} .\]

 

very short answers | Q 16 | Page 33

Write a unit vector perpendicular to \[\hat{ i } + \hat{ j }  \text{ and }  \hat{ j }  + \hat{ k } .\]

 

very short answers | Q 17 | Page 33

If \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = 144\]  and \[\left| \vec{a} \right| = 4,\]  find \[\left| \vec{b} \right|\] . 

 
 

 

very short answers | Q 18 | Page 33

If \[\vec{r} = x \hat{ i } + y \hat{ j }  + z \hat{ k }  ,\] then write the value of \[\left| \vec{r} \times \hat{ i }  \right|^2 .\]

 

 

very short answers | Q 19 | Page 33

If \[\vec{a} \text{ and }  \vec{b}\] are unit vectors such that \[\vec{a} \times \vec{b}\] is also a unit vector, find the angle between \[\vec{a} \text{ and } \vec{b}\] .

 
 

 

very short answers | Q 20 | Page 33

If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\]  write the angle between \[\vec{a} \text{ and } \vec{b} .\]

 
 

 

very short answers | Q 21 | Page 34

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then write the value of \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 .\]

 

 

very short answers | Q 22 | Page 34

If \[\vec{a}\] is a unit vector such that \[\vec{a} \times \hat{ i }  = \hat{ j }  , \text{ find }  \vec{a} . \hat{ i } \] .

 
very short answers | Q 23 | Page 34

If  \[\vec{c}\] is a unit vector perpendicular to the vectors \[\vec{a} \text{ and } \vec{b} ,\]  write another unit vector perpendicular to \[\vec{a} \text{ and }  \vec{b} .\]

 
 

 

very short answers | Q 24 | Page 34

Find the angle between two vectors \[\vec{a} \text{ and }  \vec{b}\] with magnitudes 1 and 2 respectively and when \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3} .\]

 
 
very short answers | Q 25 | Page 34

Vectors  \[\vec{a} \text{ and }  \vec{b}\] \[\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = \frac{2}{3}\text{ and } \left( \vec{a} \times \vec{b} \right)\]  is a unit vector. Write the angle between \[\vec{a} \text{ and } \vec{b}\] .

 

very short answers | Q 26 | Page 34

Find λ, if \[\left( 2 \hat{ i }  + 6 \hat{ j }  + 14 \hat{ k }  \right) \times \left( \hat{ i }  - \lambda \hat{ j } + 7 \hat{ k }  \right) = \vec{0} .\]

 
very short answers | Q 27 | Page 34

Write the value of the area of the parallelogram determined by the vectors   \[2 \hat{ i }  \text{ and } 3 \hat{ j }  .\]

 
very short answers | Q 28 | Page 34

Write the value of \[\left( \hat{ i }  \times \hat{ j }  \right) \cdot \hat{ k }  + \left( \hat{ j } + \hat{ k }  \right) \cdot \hat{ j } \]

 
very short answers | Q 29 | Page 34

Find a vector of magnitude \[\sqrt{171}\]  which is perpendicular to both of the vectors \[\vec{a} = \hat{ i } + 2 \hat{ j }  - 3 \hat{ k } \]  and  \[\vec{a} = \hat{ i } + 2 \hat{ j }  - 3 \hat{ k } \] . 

 
 
very short answers | Q 30 | Page 34

Write the number of vectors of unit length perpendicular to both the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j }  + 2 \hat{ k }  \text{ and }  \vec{b} = \hat{ j }  + \hat{ k } \] .

 
very short answers | Q 31 | Page 34

Write the angle between the vectors  \[\vec{a} \times \vec{b}\]  and  \[\vec{b} \times \vec{a}\] .

 

 

Chapter 25: Vector or Cross Product Exercise MCQ solutions [Pages 34 - 36]

MCQ | Q 1 | Page 34

If \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i }  \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k }  \right)^2 =\]

  • \[\vec{a}^2\]

     

  • \[2 \vec{a}^2\]

  • \[3 \vec{a}^2\]

     

  • \[4 \vec{a}^2\]

MCQ | Q 2 | Page 35

If \[\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}\] and \[\vec{a} \times \vec{b} = \vec{a} \times \vec{c,} \vec{a} \neq 0,\] then

  • \[\vec{b} = \vec{c}\]

  • \[\vec{b} = \vec{0}\]

  • \[\vec{b} + \vec{c} = \vec{0}\]

  • none of these

MCQ | Q 3 | Page 35

The vector \[\vec{b} = 3 \hat { i }+ 4 \hat {k }\] is to be written as the sum of a vector \[\vec{\alpha}\] parallel to \[\vec{a} = \hat {i} + \hat {j}\] and a vector \[\vec{\beta}\] perpendicular to \[\vec{a}\]. Then \[\vec{\alpha} =\]

  • \[\frac{3}{2}\left( \hat { i} + \hat {j} \right)\]

  • \[\frac{2}{3}\left( \hat {i} + \hat {j} \right)\]

  • \[\frac{1}{2}\left(\hat { i} + \hat {j} \right)\]

  • \[\frac{1}{3}\left( \hat { i} + \hat {j} \right)\]

MCQ | Q 4 | Page 35

The unit vector perpendicular to the plane passing through points \[P\left( \hat{ i } - \hat{ j }  + 2 \hat{ k }  \right), Q\left( 2 \hat{ i } - \hat{ k } \right) \text{ and }  R\left( 2 \hat{ j }  + \hat{ k }  \right)\]  is 

 
  • \[2 \hat{ i } + \hat{ j } + \hat{ k } \]

  • \[\sqrt{6}\left( 2 \hat{ i }  + \hat{ j }  + \hat{ k }  \right)\]

  • \[\frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j }  + \hat{ k }  \right)\]

  • \[\frac{1}{6}\left( 2 \hat{ i }  + \hat{ j }  + \hat{ k }  \right)\]

MCQ | Q 5 | Page 35

If \[\vec{a,} \vec{b}\] represent the diagonals of a rhombus, then

  • \[\vec{a} \times \vec{b} = \vec{0}\]

  • \[\vec{a} \cdot \vec{b} = 0\]

  • \[\vec{a} \cdot \vec{b} = 1\]

  • \[\vec{a} \times \vec{b} = \vec{a}\]

MCQ | Q 6 | Page 35

Vectors \[\vec{a} \text{ and }  \vec{b}\] are inclined at angle θ = 120°. If \[\left| \vec{a} \right| = 1, \left| \vec{b} \right| = 2,\] then  \[\left[ \left( \vec{a} + 3 \vec{b} \right) \times \left( 3 \vec{a} - \vec{b} \right) \right]^2\]  is equal to 

 
  
  • 300

  •  325

  •  275

  •  225 

MCQ | Q 7 | Page 35

If \[\vec{a} = \hat{ i }  + \hat{ j }  - \hat{ k }  , \vec{b} = - \hat{ i }  + 2\hat{ j }  + 2 \hat{ k }  \text{ and }  \vec{c} = - \hat{ i } + 2 \hat{ j }  - \hat{ k }  ,\]  then a unit vector normal to the vectors \[\vec{a} + \vec{b} \text{ and }  \vec{b} - \vec{c}\]  is

 
  • \[\hat{ i } \]

  • \[\hat{ j } \]

  • \[\hat{ k } \]

  • none of these 

MCQ | Q 8 | Page 35

A unit vector perpendicular to both \[\hat{ i }  + \hat{ j } \text{ and }  \hat{ j } + \hat{ k } \] is

 
  • \[\hat{ i }  - \hat{ j }  + \hat{ k } \]

  • \[\hat{ i }  + \hat{ j }  + \hat{ k } \] 

  • \[ \frac1 {\sqrt3}  ( \hat{ i }  + \hat{ j }  + \hat{ k } ) \] 

  • \[ \frac1 {\sqrt3}  ( \hat{ i }  - \hat{ j }  + \hat{ k } ) \] 

MCQ | Q 9 | Page 35

If \[\vec{a} = 2 \hat{ i }  - 3 \hat{ j }  - \hat{ k }  \text{ and }  \vec{b} = \hat{ i } + 4 \hat{ j }  - 2 \hat{ k 
} , \text{ then } \vec{a} \times \vec{b}\]  is

  • \[10 \hat{ i } + 2 \hat{ j }  + 11 \hat{ k } \]

  • \[10 \hat{ i }  + 3 \hat{ j }  + 11 \hat{ k } \]

  • \[10 \hat{ i } - 3 \hat{ j }  + 11 \hat{ k } \]

  • \[10 \hat{ i }  - 2 \hat{ j }  - 10 \hat{ k } \]

MCQ | Q 10 | Page 35

If \[\hat{ i }  , \hat{ j }  , \hat{ k } \] are unit vectors, then

  • \[\hat{ i }  . \hat{ j }  = 1 \]

  • \[\hat{ i }  . \hat{ i }  = 1 \]

  • \[\hat{ i }  ×  \hat{ j }  = 1 \]

  • \[\hat{ i }  ×  ( \hat{ j }   × \hat{ k} )  = 1 \]

MCQ | Q 11 | Page 35

If θ is the angle between the vectors \[2 \hat{ i }  - 2 \hat{ j}  + 4 \hat{ k }  \text{ and } 3 \hat{ i }  + \hat { j }  + 2 \hat{ k }  ,\]  then sin θ =

 
  • \[\frac{2}{3}\]

  • \[\frac{2}{\sqrt{7}}\]

  • \[\frac{\sqrt{2}}{7}\]

  • \[\sqrt{\frac{2}{7}}\] 

MCQ | Q 12 | Page 35

If \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then }  \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]

  • 6

  • 2

  • 20

  • 8

MCQ | Q 13 | Page 36

The value of \[\left( \vec{a} \times \vec{b} \right)^2\] is 

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

     

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

     

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

     

  • \[\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 - \vec{a} \cdot \vec{b}\]

     

MCQ | Q 14 | Page 36

The value of  \[\hat{ i }  \cdot \left( \hat{ j }  \times \hat{ k }  \right) + \hat{ j }  \cdot \left( \hat{ i }  \times \hat{ k }  \right) + \hat{ k }  \cdot \left( \hat{ i }  \times \hat{ j }  \right),\]  is 

  • 0

  • -1

  • 1

  • 3

MCQ | Q 15 | Page 36

If θ is the angle between any two vectors \[\vec{a} \text{ and }  \vec{b}\] , then \[\left| \vec{a} \cdot \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|\]  when θ is equal to

 
 
  • 0

  •  π/4

  • π/2

  • π

Chapter 25: Vector or Cross Product

Ex. 25.1very short answersMCQ

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) - Shaalaa.com

RD Sharma solutions for Class 12 Mathematics chapter 25 - Vector or Cross Product

RD Sharma solutions for Class 12 Maths chapter 25 (Vector or Cross Product) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics chapter 25 Vector or Cross Product are Concept of Direction Cosines, Properties of Vector Addition, Geometrical Interpretation of Scalar, Scalar Triple Product of Vectors, Vector (Or Cross) Product of Two Vectors, Scalar (Or Dot) Product of Two Vectors, Position Vector of a Point Dividing a Line Segment in a Given Ratio, Multiplication of a Vector by a Scalar, Addition of Vectors, Introduction of Vector, Magnitude and Direction of a Vector, Basic Concepts of Vector Algebra, Types of Vectors, Components of a Vector, Section formula, Vector Joining Two Points, Vectors Examples and Solutions, Projection of a Vector on a Line, Introduction of Product of Two Vectors.

Using RD Sharma Class 12 solutions Vector or Cross Product exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

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