How Do You Calculate the Parallel and Perpendicular Components of a Vector?

[hadroneater]

Homework Statement

force vector F = 700[-0.25, 0.433, 0.866].
vector A = [-4, 4, 2]
a)What is the component of F that is parallel to A?
b}And what is component of F that is perpendicular to A?

Homework Equations

$$A\bullet F = |A||F|\cos{\theta}$$
When two vectors are parallel:
$$A\bullet B = |A||B||$$
When two vectors are perpendicular
$$A\bullet B = 0$$

The Attempt at a Solution

I'm not sure if I should use the dot product to find the component but I figure that would be the simplest way to do so.
A = [-4, 4 ,2] = 6[-2/3, 2/3, 1/3]
The parallel force vector = s[-2/3, 2/3, 1/3]
That's all I have right now. If I use the dot product equation, both sides of the equation will come to the same term and cancel each other out. I think my definition for the parallel force vector is too general as s could be any scalar.

 

[JaWiB]

Note that
$$F_{parallel} = |F|\cos{\theta}$$
(this is obvious if you draw two vectors on paper and apply a little trig)

So
$$A\bullet F = |A|F_{parallel}$$
Then just use the definition of the dot product to calculate the left hand side and solve for the parallel component

 

[kerimek]

As a scientist, you are correct in using the dot product to find the component of vector F that is parallel to A. The dot product equation you provided is the correct one to use in this case. However, your solution is not complete. Let's look at the steps you need to take:

a) To find the component of F that is parallel to A, we need to find the scalar s that will give us the parallel force vector. Using the dot product equation, we have:

A•F = |A||F|cosθ

Substituting the values given in the problem, we have:

[-4, 4, 2]•[700(-0.25), 700(0.433), 700(0.866)] = |[-4, 4, 2]||[700(-0.25), 700(0.433), 700(0.866)]|cosθ

Simplifying, we get:

-2800(-0.25) + 2800(0.433) + 1400(0.866) = 6√21√(-0.25^2 + 0.433^2 + 0.866^2)cosθ

This reduces to:

-700 + 1212 + 1212 = 6√21√(0.75)cosθ

Solving for cosθ, we get:

cosθ = 0.5

Knowing that cosθ = adjacent/hypotenuse, we can set up a proportion to solve for the scalar s:

0.5 = s/700

Solving for s, we get:

s = 350

Therefore, the component of F that is parallel to A is 350[-0.25, 0.433, 0.866].

b) To find the component of F that is perpendicular to A, we can use the Pythagorean theorem. Since we already know the magnitude of the parallel component (350), we can find the magnitude of the perpendicular component by subtracting the magnitude of the parallel component from the magnitude of F:

|F| = √(700^2 + 700^2 + 700^2) = 700√3

|F|perpendicular = √(700√3)^2 - (350)^2 = √(2450000) - 122500 =