I have a problem where i need to find an vector $\vec{p}$, and all i have is angles with Ox and Oy, and I know that angle with Oz is obtuse. Professor in our materials used the formula ${\cos^2{\alpha}}+{\cos^2{\beta}}+{\cos^2{\gamma}}=1$ with $\alpha, \beta, \gamma$ being angles between the vector $\vec{p}$ and Ox, Oy Oz respectivly. Another thing i didn't understand why unit vector $\vec{p_0}$ was set to ($\cos{\alpha}, \cos{\beta}, \cos{\gamma}$) when vector $\vec{p}$ was being being calculated. Where was this formula derived from, and why is unit vector as is?


1 Answer 1


If $u = (x, y, z)$ is a unit vector, and $i,j,k$ are the unit vector along $Ox, Oy, Oz$, then

$\cos \alpha = u \cdot i = x $

$\cos \beta = u \cdot j = y $

$\cos \gamma = u \cdot k = z $

Since $x^2 + y^2 + z^2 = 1 $ because $u$ is a unit vector, it follows that

$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 $


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