Part 4: Direction angles and Direction Cosines

Given a vector v, let us let a denote the angle between v and the x-axis, let us let b denote the angle between v and the y-axis, and let us let g denote the angle between v and the z-axis.

(Click and drag arrow's endpoint. Angles are in degrees.)

The angles a, b, and g are called the direction angles for v. 

If v = á a,b,c ñ = ai+bj+ck, then

v·i = a i·i+b i·j+c i·k = a

so that a = ||v|| || i|| cos( a) = ||v|| cos( a).  Similarly, b = v·j and c = v·k, so that

a = ||v|| cos( a) ,  b = ||v|| cos( b) ,  c = ||v|| cos( g)

(7)

When combined with v = á a,b,c ñ , this leads to

v = ||v||    á cos( a) ,cos( b) ,cos( g) ñ

That is, the direction vector for v is the unit vector given by

u = á cos( a) ,cos( b) ,cos( g) ñ

The components of u are the direction cosines for v.        

EXAMPLE 8    Determine the direction cosines and direction angles for

v = á 1,1,Ö2 ñ

Solution: Since ||v|| ² = 1+1+2 = 4, the magnitude of v is ||v|| = 2. Equation (7) implies that

1 = 2cos( a) 1 = 2cos( b) Ö2 = 2cos( g) cos( a) = 1 2 cos( b) = 1 2 cos( g) =  2 Ö2 a =    p 3 b =    p 3 g =    p 4

as is shown below:

(Angles are in degrees.)