Question

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?

Answer 1

`|vec F_A| = (9 sqrt(17))/(17)`

Explanation:

We can use dot products!
In general,
`|vec F_A| * |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 = 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 * -3 + 5 * 2 + -2 * -1 = 9`
`vec A cdot vec A = (-3)^2 + 2^2 + (-2)^2 = 17 = |vec A|^2`

hence
`|vec F_A| * sqrt(17) = 9 implies |vec F_A| = (9 sqrt(17))/(17)`

our final answer.