Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given the position vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k}\] and \[- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\]
Let \[\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}\] and \[\vec{b} = - 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\]
Then,
\[\vec{b} = - 4 \hat{i} + 6 \hat{j} - 8 \hat{k} \]
\[ = - 2\left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right)\]
\[ = - 2 \vec{a} \]
Hence, \[\vec{a} , \vec{b}\] are collinear.
APPEARS IN
संबंधित प्रश्न
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.
Write the position vector of the point which divides the join of points with position vectors `3veca-2vecb and 2veca+3vecb` in the ratio 2 : 1.
Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1
If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.
Represent graphically a displacement of 40 km, 30° east of north.
Classify the following as scalar and vector quantity.
Time period
Show that the vector `hati + hatj + hatk` is equally inclined to the axes OX, OY, and OZ.
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of the x-axis.
Find the value of x for which `x(hati + hatj + hatk)` is a unit vector.
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.
Find the angle between the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - 2 \hat{k}\]
If \[\hat{ a } \text{ and } \hat{b }\] are unit vectors inclined at an angle θ, prove that
\[\tan\frac{\theta}{2} = \frac{\left| \hat{a} -\hat{b} \right|}{\left| \hat{a} + \hat{b} \right|}\]
If \[\vec{a,} \vec{b,} \vec{c}\] are three mutually perpendicular unit vectors, then prove that \[\left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{3}\]
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 \[\vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] 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.
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?
Find the vector from the origin O to the centroid of the triangle whose vertices are (1, −1, 2), (2, 1, 3) and (−1, 2, −1).
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.
If \[\vec{a} \times \vec{b} = \vec{c} \times \vec{d} \text { and } \vec{a} \times \vec{c} = \vec{b} \times \vec{d}\] , show that \[\vec{a} - \vec{d}\] is parallel to \[\vec{b} - \vec{c}\] where \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .
If A, B, C, D are the points with position vectors `hat"i" + hat"j" - hat"k", 2hat"i" - hat"j" + 3hat"k", 2hat"i" - 3hat"k", 3hat"i" - 2hat"j" + hat"k"`, respectively, find the projection of `vec"AB"` along `vec"CD"`.
The altitude through vertex C of a triangle ABC, with position vectors of vertices `veca, vecb, vecc` respectively is:
Area of rectangle having vertices A, B, C and D will position vector `(- hati + 1/2hatj + 4hatk), (hati + 1/2hatj + 4hatk) (hati - 1/2hatj + 4hatk)` and `(-hati - 1/2hatj + 4hatk)` is
Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2
If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.
Geometrically, a vector is represented by which of the following?
In the vector \[\vec{AB}\], what are A and B respectively?
Which notation can be used for the magnitude of vector \[\vec{AB}\]?
In three-dimensional geometry, what is the vector drawn from \[O(0,0,0)\] to \[P(x,y,z)\] called?
