Why is cos^2 a + cos^2 b + cos^2 c =1 in vectors when a, b and c are angles which a vector OP makes with the O

Question

Why is cos^2 a + cos^2 b + cos^2 c =1 in vectors when a, b and c are angles which a vector OP makes with the Ox, Oy and Oz respectively?
Coordinates of p are (x, y , z).

Please explain clearly!

Answer 1

To get the angle drop a perpendicular from the top of the vector to the x-axis, which will hit the Ox at x. Now you have a right triangle with two lines forming the angle. The hypothenuse is the vector. Its length ist given by:
l = sqrt(x²+y²+z²)
The other cathete is line on Ox with the length x.
In a right triangle the angle between a cathete and the hypothenuse is given by:
cosine(angle) equals length of cathete divided by length of hypothenuse.
In this case:
cos(a) = x/l=x/sqrt(x²+y²+z²)
Anologous calculation for the other axes give:
cos(b) = y/l=y/sqrt(x²+y²+z²)
cos(c) = z/l=z/sqrt(x²+y²+z²)

Therefore:
cos²(a]+cos²(b)+cos²(c)
=x²/l²+y²/l²+z²/l²
=(x²+y²+z²)/l²
=(x²+y²+z²)/(x²+y²+z²)
= 1
q.e.d.