Find the value of λ for which and are perpendicular, where

i. and

ii. and

iii. and

iv. and

i. If and show that is perpendicular to

ii. If and then show that and are orthogonal.

If and then find the value of λ so that and are orthogonal vectors.

Show that the vectors

and

are mutually perpendicular unit vectors.

Let and

Find a vector which is perpendicular to both and and is such that

Let and

Find the projection of (i) on and (ii) on

Find the projection of in the direction of

Write the projection of vector along the vector

i. Find the projection of on if and

ii. Write the projection of the vector on the vector

Find the angle between the vectors and when

i. and

ii. and

iii. and

If and then calculate the angle between and

If is a unit vector such that find

Find the angles which the vector makes with the coordinate axes.

Show that the vector is equally inclined to the coordinate axes.

Find a vector of magnitude making an angle π/4 with x - axis, π/2 with y - axis and an acute angle θ with z - axis.

Find the angle between and if and

Express the vector as sum of two vectors such that one is parallel to the vector and the other is perpendicular to

Prove that where and

If and find the angle between and

Find the angle between and when

i. and

ii. and

If and find

If and find and

If and are unit vectors inclined at an angle θ then prove that:

i.

ii.

The dot products of a vector with the vector and are 0, 5 and 8 respectively. Find the vector.

If and the coordinates of A are (0, - 2, - 1), find the coordinates of B.

If A(2, 3, 4), B(5, 4, - 1), C(3, 6, 2) and D(1, 2, 0) be four points, show that is perpendicular to

Find the value of λ for which the vectors and are perpendicular to each other.

Show that the vectors and form a right - angled triangle.

Three vertices of a triangle are A(0, - 1, - 2), B(3, 1, 4) and C(5, 7, 1). Show that it is a right - angled triangle. Also, find its other two angles.

If the position vectors of the vertices
A, B and C of a ∆ABC be (1, 2, 3), ( - 1, 0, 0) and (0, 1, 2) respectively then find ∠ABC.

If and are two unit vectors such that find

If and are two vectors such that then prove that vector is perpendicular to the vector

If and then express in the form where and