Question

We are given that a force of 60 N is exerted at an angle of 34°. Therefore, the force vector F can be represented as the hypotenuse of a right triangle with an acute angle 34°. The length of the hypotenuse would be
|F| = ?.

Answer 1

Would this simply be recognizing that the x-component is 60cos(34) and that the y-component is 60sin(34) then using pythagorean theorem to find the maginitude of the force vector?

Answer 2

It's a trick question. The hypotenuse represents the vector itself, the other two sides of the triangle are the x and y components of this vector. The component of force in the x direction is Fcos(34 deg), and the component of force in the y direction is Fsin(34 deg). Written in vector form this says:

Force vector = ( Fcos(34 deg) , Fsin(34 deg))

Like I said, the force vector is the hypotenuse and the other two sides have lengths of Fcos(34 deg) and Fsin(34 deg). Therefore we can get the "magnitude" (length) of this vector by using the Pythagorean theorem:

|Hypotenuse|^2 = side^2 + side^2

|Force vector|^2 = (F^2)cos^2(34 deg) + (F^2)sin^2(34 deg)

or:

|Force vector|^2 = F^2(cos^2(34) + sin^2(34))

But, from the Pythagorean theorem we know that cos^2 + sin^2 = 1. So this whole complicated thing is reduced to:

|Force vector|^2 = F^2, or

|Force vector| = F

So wait? All of that complicated math just says that the answer would just be the force they told you at the beginning: 60 N, and it didn't even matter what angle they gave you because it would have worked out the same way no matter what. Actually, by understanding the basics of how vectors work, you don't need to do ANY MATH AT ALL to answer this question.

Answer 3

I thought that same thing to begin with, but then why they would even ask such a trivial question, and doubted my reasoning. I can't believe I doubted it!!!!!arghhhhhh

Answer 4

Yes I second guessed a little bit too, I was like wait is this a trick question? Haha but the math never fails