1) Suppose F is acting vertically downward on an object sitting on a plane that is inclined at an angle of 45 degrees to the horizontal. Express this force as a sum acting parallel to the plane and one acting perpendicular to it.
F = || F|| cos (?) + ||F|| sin (?)
2) an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion.
I thought maybe the cosine law would work because the direction of movement given and the force vector given aren't parallel so the perpendicular force wouldn't be orthogonal to both.
but I couldn't get that to work
not sure if I am wording this right but the angle between the normal and weight is equal to the angle of the incline because it's basically a rotation of 90 degrees. I know thats not worded correctly, those angles can form a congruent triangle to the triangle formed by the incline
i'll give you what the book has the answer for (1)
Make sure you know this analysis back and front because you will see it over and over.
where theta is the angle between the force and the direction of motion. So find alpha from this.
For the perpendicular part use the dot product again. This time, though, you need to use the dot product to find a vector at right angles to the direction of motion. So
(because cos(90) = 0.)
This will give you the value of b in terms of a, which you can arbitrarily choose to be 1. Now do the same as above with your new vector perpendicular to the direction of the motion.