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

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

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:

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

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

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

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

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?

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Unit Vectors concept. You can view video lessons to learn Unit Vectors. Or if you need more Unit Vectors practice, you can also practice Unit Vectors practice problems.