Thread: Vectors in 3 Dimensions

Show that if a vector in 3 dimensions makes angles a, b, c respectively with the x, y, z axes, then cos^2(a) + cos^2(b) + cos^2(c) = 1.

How do I tackle this question? Thanks.

Originally Posted by classicstrings

Show that if a vector in 3 dimensions makes angles a, b, c respectively with the x, y, z axes, then cos^2(a) + cos^2(b) + cos^2(c) = 1.

How do I tackle this question? Thanks.

Hello, classicstrings,

let s be the length of the vector v.

Then v can be expressed by its coordinates:

v = (s*cos(a), s*cos(b), s*cos(c)= s * (cos(a), cos(b), cos(c))

Now calculate the length of v:

|v| = s = sqrt(s²*cos²(a) + s²*cos²(b) + s²*cos²(c)) = s * sqrt(cos²(a) + cos²(b) + cos²(c))

Divide both sides of this equation by s and you'll get:

1 = sqrt(cos²(a) + cos²(b) + cos²(c))

Squaring both sides of the last equation will give the desired result.

EB

Thanks a lot!