Show that the vector i ^ + j ^ + k ^ is equally inclined with the axes O X , O Y and O Z .

Let r → = i ^ + j ^ + k ^ and it make an angle r → = i ^ + j ^ + k ^ with O X , O Y , O Z respectively. Then its direction cosines are cos α , cos β and cos γ . Therefore, direction cosines of r → are 1 1 2 + 1 2 + 1 2 , 1 1 2 + 1 2 + 1 2 , 1 1 2 + 1 2 + 1 2 ⇒ 1 3 , 1 3 , 1 3 Thus, cos α = 1 3 , cos β = 1 3 and cos γ = 1 3 ⇒ α = β = γ Hence, all are equally inclined with the coordinate axis.

If a unit vector a → makes an angle π 3 with i ^ , π 4 with j ^ and an acute angle θ with k ^ , then find θ and hence, the components of a → .

The Direction cosines of vector a → are l = cos π 3 = 1 2 , m = cos π 4 = 1 2 , n = cos θ Therefore, l 2 + m 2 + n 2 = 1 ⇒ 1 4 + 1 2 + cos 2 θ = 1 ⇒ cos θ = 1 2 ⇒ θ = π 3 [ ∵ a → makes acute angle with k ^ ] Since, a → is the unit vector. ∴ a → = l i ^ + m j ^ + n k ^ ⇒ a → = 1 2 i ^ + 1 2 j ^ + 1 2 k ^ Hence, components of a → are 1 2 i ^ + 1 2 j ^ + 1 2 k ^ .

Find a vector r → of magnitude 3 2 units which makes an angle of π 4 and π 4 with y and z - axes respectively.

Suppose vector r → makes an angle α with the x - axis. Let l , m , n be the direction cosines of r → . Then, l = cos α , m = cos π 4 = 1 2 , n = cos π 2 = 0 Now, l 2 + m 2 + n 2 = 1 ⇒ cos 2 α + 1 2 + 0 = 1 ⇒ cos α = ± 1 2 We know that r → = r → l i ^ + m j ^ + n k ^ ∴ r → = 3 2 ± 1 2 i ^ + 1 2 j + 0 k ^ r → = 3 2 ⇒ r → = ± 3 i ^ + 3 j ^ .