3D Vector defined by 3 angles trigonometry components

Question

What I'm looking for is the trigonomery equations to calculate the x, y and z components of a 3D vector. What I mean:

The counterpart formulas for a 2D vector defined by 1 angle:

$x = \cos(\alpha)$

$z = \sin(\alpha)$

The counterpart for a 3D vector defined by 2 angles:

$x = \cos(\alpha) \cos(\beta)$

$z = \sin(\alpha) \cos(\beta)$

$y = \sin(\beta)$

So what I need is something along the lines of:

$x = \cos(\alpha) \cos(\beta) f(\gamma)$

$z = \sin(\alpha) \cos(\beta) g(\gamma)$

$y = \sin(\beta) h(\gamma)$

where $f(\gamma),g(\gamma),h(\gamma)$ are some functions of $\gamma$.

Answer 1

You can use the three angles between the vector $\mathbf{v}$ and the coordiante axes, defined by the direction cosines:

$$ \cos \alpha=\dfrac{\langle \mathbf{v},\mathbf{i} \rangle}{|\mathbf{v}|} $$

$$ \cos \beta=\dfrac{\langle \mathbf{v},\mathbf{j} \rangle}{|\mathbf{v}|} $$

$$ \cos \gamma=\dfrac{\langle \mathbf{v},\mathbf{k} \rangle}{|\mathbf{v}|} $$ so that you have: $$ \mathbf{v}=\begin{bmatrix}x\\y\\z \end {bmatrix}= |\mathbf{v}|\begin{bmatrix}\cos \alpha\\\cos \beta\\\cos \gamma \end {bmatrix} $$