Relation between Direction Cosines

Theorem

Let $\LL$ be a line embedded in a Cartesian $3$-space.

Let $l$, $m$ and $n$ be the direction cosines of $\LL$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively.

Then:

$l^2 + m^2 + n^2 = 1$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): direction angles
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): direction angles