Question

If two forces of 5 lb. and 8 lb. are acting on the same point are equivalent to a single force of 10 lb., find the angle between the given forces, and the angle between the resultant and the larger force.

I am not allowed to use law of cosines I must use vectors.

Answer 1

set up the forces so that the 8lb force is along the x-axis. Then it is <8,0>

Now the 5lb force is at an angle θ such that
(8+5cosθ)^2 + (0+5sinθ)^2 = 10^2
64+80cosθ+25cos^2θ + 25sin^2θ = 100
64+80cosθ+25 = 100
80cosθ = 11
cosθ = 11/80

Note that this is exactly equivalent to using the law of cosines (go ahead, try it -- you'll see the exact same coefficients popping up.

Note that θ is the answer to the 1st part. Now use that to find the 2nd part. If it doesn't seem obvious, draw the diagram; that should help.

Answer 2

okay for the first part I got 82.079 degrees