Problem: Let V = 20.5 i + 22.5 j − 13.5 k.(a) Find the magnitude of V.(b) What angle does this vector make with the x-, y-, and z-axes?

Remember that the unit-vector form is a vector equation that relates a vector to its x, y, and z components:

$\boxed{\stackrel{\mathbf{\rightharpoonup }}{\mathit{A}}\mathbf{=}{\mathit{A}}_{\mathit{x}}\mathbf{ }\hat{\mathbf{i}}\mathbf{+}{\mathit{A}}_{\mathit{y}}\mathbf{ }\hat{\mathbf{j}}\mathbf{+}{\mathit{A}}_{\mathit{z}}\mathbf{ }\hat{\mathbf{k}}}$

Anytime we're asked to relate the magnitude and direction of a three-dimensional vector to its components, there are four basic equations we might use:

$\boxed{\mathbf{\left|}\stackrel{\mathbf{\rightharpoonup }}{\mathit{A}}\mathbf{\right|}\mathbf{=}\sqrt{{{\mathit{A}}_{\mathit{x}}}^{\mathbf{2}}\mathbf{+}{{\mathit{A}}_{\mathit{y}}}^{\mathbf{2}}\mathbf{+}{{\mathit{A}}_{\mathit{z}}}^{\mathbf{2}}}}$ (1)

$\boxed{{\mathit{A}}_{\mathit{x}}\mathbf{=}\mathbf{\left|}\stackrel{\mathbf{\rightharpoonup }}{\mathit{A}}\mathbf{\right|}\mathbf{ }\mathbf{cos}\mathbf{ }\mathit{\alpha }}$ (2)

$\boxed{{\mathit{A}}_{\mathit{y}}\mathbf{=}\mathbf{\left|}\stackrel{\mathbf{\rightharpoonup }}{\mathit{A}}\mathbf{\right|}\mathbf{ }\mathbf{cos}\mathbf{ }\mathit{\beta }}$   (3)

$\boxed{{\mathit{A}}_{\mathit{z}}\mathbf{=}\mathbf{\left|}\stackrel{\mathbf{\rightharpoonup }}{\mathit{A}}\mathbf{\right|}\mathbf{ }\mathbf{cos}\mathbf{ }\mathit{\gamma }}$   (4)