# Magnitude of projection force

1. Sep 29, 2016

### Robb

1. The problem statement, all variables and given/known data

Determine the magnitude of the projection of force F = 700 N along the u axis

2. Relevant equations

3. The attempt at a solution

A(-2, 4, 4)
r(AO) = 2i -4j - 4k
r(AO mag)= 6
u(AO) = 1/3i - 2/3j - 2/3k
F(AO) = 233.3333i - 466.6667j - 466.6667k

I'm not sure where to go from here. The only info I can figure about u is the 30 degree angle between it and the y axis.

2. Sep 29, 2016

### Staff: Mentor

This is a vector problem. Do you know how to find the projection of one vector along the direction of another?

3. Sep 29, 2016

### Robb

F dotted with u(AO)

4. Sep 29, 2016

### Staff: Mentor

Hmm. Don't you mean F dotted with u? Note that F lies along the direction OA. So you'll need to construct a vector for F and another for the u unit vector. That 30° angle should come in handy for the latter.

5. Sep 29, 2016

### Robb

F = 233.3333i - 466.6667j - 466.6667k

I guess I'm just not seeing the u unit vector. And, yes, that is what i meant.

6. Sep 29, 2016

### Staff: Mentor

I think you need to check the signs of your F components. It looks to me from the diagram that F is pointing towards the negative x-axis, positive y-axis, and positive z-axis.

The u unit vector should lie along the u-axis. Use a bit of trig to find the x and y components. It appears to lie in the x-y plane...

7. Sep 29, 2016

### Robb

u(x) = 4/cos30 = 4.62
u(y) = sqrt(4.62^2 - 4^2) = 2.31
u = .5i +.866j
F(u) = (-233.3333i)(.5) + (466.6667)(.866) = 287.5

They want the answer to two significant figures

8. Sep 29, 2016

### Staff: Mentor

The simplest approach to finding a unit vector is to imagine that it sits within a unit circle, and then the sine and cosine of a suitable angle will give your the components directly. The magnitude of any vector with unit length is always unity, just like the radius of the unit circle.

The axis u is indicated as having an angle of 30° with respect to the y-axis. So you should expect the y-component to be the cosine of the angle and the x-component to be the sine of the angle. You seem to have found the opposite (and I don't really understand where your value of 4 came from... there's nothing indicated in the figure that ascribes a dimension of 4 in relation to the u-axis).

This is what you have to work with:

9. Sep 29, 2016

### Robb

Yep, I understand that. I was grabbing at straws because I'm not sure how to get values for u.

10. Sep 29, 2016

### Staff: Mentor

Okay, so are you good to go now?

11. Sep 29, 2016

### Robb

So, I have u(y) = ucos30 and u(x) = usin30
I guess where I'm lost in all of this is what is u? I need to use the dot product to solve but without a value for the above components how can I do that. Feeling a little blind here.

12. Sep 29, 2016

### Staff: Mentor

u is a unit vector along the positive u-axis. Refer to the diagram that I posted.

All you need in order to find a projection along a given direction is to dot a given vector with a unit vector in the desired direction. Works with any unit vector; You can try it with your F vector and the unit vectors for the x,y, and z axis if you like. Take the dot product of F with any of the axes unit vectors and you should "extract" that component from the F vector.

13. Sep 30, 2016

### Robb

usin30 + ucos30 = 1
u = .7214
u(x) = .3607
u(y) = .6248
F dot u = 209
To two sig figs F(u) = 210

14. Sep 30, 2016

### Staff: Mentor

No, how do vector components add?
u can't be both a scalar and a vector. Besides, the sine and cosines all by themselves satisfy the requirement of unit vector components. $sin^2 + cos^2 = 1$. So just use the sine and cosine as the components of u.
You'll need to re-do that with the fixed u.

15. Sep 30, 2016

### Robb

yeah, that wasn't real smart.

F dot u = (-233.3333)(.5) + (466.6667)(.8661) = 287.5

two sig figs = 290

16. Sep 30, 2016

### Staff: Mentor

You've mixed up the components of u again. Look at the diagram in post #8. Is the sine of the angle along the y-axis or the x-axis?

17. Sep 30, 2016

### Robb

sin=x
cos=y

18. Sep 30, 2016

### Staff: Mentor

Right. And sin(30°) = 1/2, while cos(30°) = √3 / 2.

19. Sep 30, 2016

### Robb

I agree: (-233.333)(.5) + (466.6667)(.8661) = 287.5 or 290

20. Sep 30, 2016

### Staff: Mentor

D'oh! I apologize. I misread and jumped without engaging my brain

You have indeed got u sorted out now. So your result is correct