two forces F1 and F2 of magnitudes PN an QN acting in the directions I-2j and 4i+3j respectively. Given that the resultant of F1 and F2 is F, show that P=16 and find Q where F=(48i+14j)N.

Make a sketch, this will greatly help you.

From the sketch, you know that:

|\vec{F}|^2 = |\vec{P}|^2 + |\vec{Q}|^2 - 2|\vec{P}||\vec{Q}|\cos(\angle{OQF})

Also that:

\frac{\sin(\angle{OQF})}{|\vec{F}|} = \frac{\sin(\angle{FOQ})}{|\vec{P}|} = \frac{\sin(\angle{QFO})}{|\vec{Q}|}

It's simultaneous equations :)

It doesn't work out to 16. It works out to P=8\sqrt5.

To start with, you have

P\frac{i-2j}{|i-2j|}+Q\frac{4i+3j}{|4i+3j|} = 48i+14j

or

\frac{P}{sqrt 5}(i-2j)+\frac{Q}{5}(4i+3j) = 48i+14j

From that, you can just separate into two equations (since i and j are orthogonal to each other, they're linearly independent):

The i terms: \frac{P}{sqrt 5}+\frac{4Q}{5} = 48

The j terms: \frac{-2P}{sqrt 5}+\frac{3Q}{5} = 14

We can solve by elimination by doubling the first equation and adding the two together to get:

\frac{11Q}{5}=110

Q=50

Plugging that back into the first equation, we get

\frac{P}{sqrt 5}+\frac{4 \cdot 50}{5} = 48

\frac{P}{sqrt 5} = 8

P = 8{sqrt 5}\;\approx\;17.9

Evejack, is there a typo in your question? If F1 points in the (i-2j) direction, there's no way both P and F can simultaneously have rational components.

As I read this question they have given you that P = 16 Newtons of force acting in the direction given by I-2j. Since this sets the direction of the force, P, you get that P is acting at an angle of -63.43 degrees. The angle is found by converting the unit direction vector of I-2j to its polar equalivent, [email protected] degrees. I am assuming no affect for the unit vector here except to give direction, so the P vector would be given as, [email protected] degrees. Then adding the Q vector to this we get a resultant force vector F of 48i+14j = [email protected] degrees. Now if you subtract the two vectors from each other you get:
F - P = Q; (48i+14j) - (7.16i-14.31j) = 40.84i+28.31j = [email protected] degrees = QN

Newton1Law, I agree with your math, but my interpretation of the question is different. To me the question pretty clearly says to show that P=16, not to find Q if P=16. Besides, it's crystal clear that Q is supposed to be in the 4i+3j direction (36.9 degrees if you want to convert to polar coordinates). Your answer for Q is not in that direction. My only conclusion is that the question must be invalid as written.

Yes it had a typo. Thanks the answers are correct. Thanks jcaron2's

yes it is a simultaneous equation. Thanks man.

You're welcome :)