Why is the sum of cosines of angles defining a point

Question

Consider a point P in 3D space which makes an angle $\alpha$ to the $x$-axis an angle $\beta$ to the $y$-axis and an angle $\gamma$ to the $z$-axis. The sum of the squares of the cosines of these angles will equal 1: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Why is this?

Answer 1

Let's define a vector $\vec{v} = (x,y,z)$

$$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$ $$\left({x\over |\vec{v}|}\right)^2 + \left({y\over |\vec{v}|}\right)^2 + \left({z\over |\vec{v}|}\right)^2 = 1$$

$$\frac{x^2+y^2+z^2}{|\vec{v}|^2}=1$$ And you got the pythagoras theorem $$x^2+y^2+z^2=|\vec{v}|^2$$