The direction cosines of a vector $v$ are the cosine of the angles between $v$ and the coordinate axes.
Consider a vector $v \in \mathbb{R}^3$ and let $x, y, z$ represent each of the coordinate axes. Let $\hat{x}, \hat{y},\hat{z}$ be the unit vectors along each axis and let the angle between each axis and $v$ be $\theta_x,\theta_y, \theta_z$ respectively. The direction cosines are:
$\cos(\theta_x) = \displaystyle\frac{v \cdot \hat{x}}{|v|}$
$\cos(\theta_y) = \displaystyle\frac{v \cdot \hat{y}}{|v|}$
$\cos(\theta_z) = \displaystyle\frac{v \cdot \hat{z}}{|v|}$
Do the direction cosines generalize to higher dimensions? For example if we are in $\mathbb{R}^5$ with axes $p,q,r,s,t$ and directional unit vectors $\hat{p}, \hat{q},\hat{r}, \hat{s},\hat{t}$, can we consider a vector $w \in \mathbb{R}^5$ to make angles $\theta_p,\theta_q, \theta_r, \theta_s, \theta_t$ with these axes and have the direction cosines
$\cos(\theta_p) = \displaystyle\frac{w \cdot \hat{p}}{|w|}$
$\cos(\theta_q) = \displaystyle\frac{w \cdot \hat{q}}{|w|}$
$\cos(\theta_r) = \displaystyle\frac{w \cdot \hat{r}}{|w|}$
$\cos(\theta_w) = \displaystyle\frac{w \cdot \hat{s}}{|w|}$
$\cos(\theta_t) = \displaystyle\frac{w \cdot \hat{t}}{|w|}$
where the angles $\theta_p,\theta_q, \theta_r, \theta_s, \theta_t$ are angles in some higher-dimensional coordinate system?
References:
Weisstein, Eric W. "Direction Cosine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirectionCosine.html