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?

1 Answer
Aug 8, 2018

Answer:

#|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.