Direction cosines

If vector makes angle α, β and γ with x, y and z axes respectively, then

cosα=

where cos α, cos β and cos γ are called direction cosines of the vector. The unit vector in the direction of is

=

Illustration 10. If and, find the vector having the same magnitude as and parallel to.

Solution: Magnitude of = = 5

And magnitude of = 25

Now a unit vector parallel to =

∴ The vector having the same magnitude as and parallel to

= 25

Illustration 11. What is the displacement of a point on a wheel initially in contact with the ground when the wheel rolls froward through half a revolution. Given radius of wheel is R.

Solution: Displacement =

=

tan θ =

∴ θ = tan−1