# A force is given as F=x+5y-z. How do you calculate the magnitude of the force in the direction of the vector A=-3x+2y-2z?

Aug 8, 2018

$| {\vec{F}}_{A} | = \frac{9 \sqrt{17}}{17}$

#### Explanation:

We can use dot products!
In general,
$| {\vec{F}}_{A} | \cdot | \vec{A} | = \vec{F} \cdot \vec{A}$
where ${\vec{F}}_{A}$ is the component of the force along $\vec{A}$.

This may be easier to see if we just think of normalizing $\vec{A}$ s.t. $\vec{a} = \frac{\vec{A}}{|} \vec{A} |$. Hence we can easily see that,

$| {\vec{F}}_{a} | = \vec{F} \cdot \vec{a}$

Using the first equation, we can easily calculate both known values:
$\vec{F} \cdot \vec{A} = 1 \cdot - 3 + 5 \cdot 2 + - 2 \cdot - 1 = 9$
$\vec{A} \cdot \vec{A} = {\left(- 3\right)}^{2} + {2}^{2} + {\left(- 2\right)}^{2} = 17 = | \vec{A} {|}^{2}$

hence
$| {\vec{F}}_{A} | \cdot \sqrt{17} = 9 \implies | {\vec{F}}_{A} | = \frac{9 \sqrt{17}}{17}$