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JoeXIII'007
09-05-2007, 08:31 AM
ok... this problem is driving me nuts and I have till tomorrow (9/6/07) to figure it out.

It involves 3 dimensional vector math and dot products.

The task is to "find the magnitude of the projection of the load supporting cable OA (O being origin) onto the positive z-axis" as shown in a figure. The figure displays 3 vectors A, B, and C plus a 4th force vector going down the negative z-axis which is shown to have a force of 1000kg.

Points and their coordinates are as follows:

OA.) (10,5,20)
OB.) (-5,-5,20)
OC.) (5,-5,20)

From there, I am a bit stuck. I know the formula of any projection of u onto v... but it doesnt seem to me it would work in this case, it just doesnt seem right. Trig appears to me to be involved but how to apply it sort of stumps me for some reason.

Any help (hints, strategies, etc.) would be helpful and greatly appreciated... thanks!

-Joe

Liz Smith
09-05-2007, 08:57 AM
I'm running out the door, but from my brief look at you question, my advice to you is to make sure you are using the correct angle in your dot product equation. Remember, you're working in 3D, so you have to do a little more calculations to find the angle between the vector and the axis.

Jack Jones
09-05-2007, 09:55 AM
Hint 1: This is a one dimentional problem.
Hint 2: Normalize the three vectors.
Hint 3: Ignore the x and y components
Hint 4: Think Ratio

Kevin Sevcik
09-05-2007, 12:13 PM
Joe,

Do you know what form the expected answer should take? If it's simply looking for the ratio of the magnitude of OA to the projection, then that's pretty easy and Jack's advice applies.

If they're expecting you to calculate an actual load on OA, then that's a little trickier. In that case it turns into a system of 3 equations. On that track, the idea is that the loads on OA, OB, and OC add up to your load, but in the opposite direction. So FOA + FOB + FOC = (0,0,1000). The trick there is, again, to nomalize (turn into a unit vector) OA, OB, OC. Then you can multiply those unit vectors by a magnitude and end up with a system of 3 equations.

JoeXIII'007
09-05-2007, 01:09 PM
Joe,

Do you know what form the expected answer should take? If it's simply looking for the ratio of the magnitude of OA to the projection, then that's pretty easy and Jack's advice applies.

If they're expecting you to calculate an actual load on OA, then that's a little trickier. In that case it turns into a system of 3 equations. On that track, the idea is that the loads on OA, OB, and OC add up to your load, but in the opposite direction. So FOA + FOB + FOC = (0,0,1000). The trick there is, again, to nomalize (turn into a unit vector) OA, OB, OC. Then you can multiply those unit vectors by a magnitude and end up with a system of 3 equations.

All I know is that they want the magnitude of the projection of OA onto the positive z axis... I dont think they want the load. So I am assuming that they want how much OA is pulling up on the Z axis. In other words, what part does it have in the system of 3 cables since it isnt a single cable pulling up on the z axis.

The exact question from the book is in quotes in the original post... admittingly it is pretty inspecific.

Kevin Sevcik
09-05-2007, 01:28 PM
In that case, I suspect you're looking at my second paragraph of hints, then.

JoeXIII'007
09-06-2007, 12:07 AM
Well... here's what I got so far:

I pretty much focused on OA... I did normalize the vector equation to get

[2/sqrt(21)] i + [1/sqrt(21)] j + [4/sqrt(21] k *This is after simplification, the magnitude of OA is 5sqrt(21)*

Realizing this will create ratios of force, and that the magnitude of the normal equation is 1, I then put this through the projection formula (http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html):

projv u = (u.v/||v||^2)v

Letting u = OA and v = the 1000kg vector... the x and y components logically no longer matter and its projection comes out to be

(4sqrt(21))/21 k

Then finding the magnitude of that projection resulted in:

16/21


And that is how the cookie crumbled for me... does that seem right??? :confused:

I do like to learn different routes of logic, and vector math in calc 3 has definitely introduced me to that... so I really do appreciate the hints and everything. Helps me out a lot. :cool:

_joe

Update: thinking about it... the 16/21 had to be its portion of the load... so the actual load bearing capacity i came up with was ~762... but i still dont know if its right...

Kevin Sevcik
09-06-2007, 09:20 PM
Joe,

Why don't you sanity check the answer you got there? Apply the same procedure to OB and OC and see if your answers add up to 1000. If they don't... well then you're barking up the wrong tree.