Solutions to Dot Product Exercises

1. You can't prove that the dot product is associative because v · w · x doesn't make sense: either way you evaluate it, you should have to take the dot product of a vector and a scalar, which is impossible.
2. (1,2,3) · (4,5,6) = 4 + 10 + 18 = 32.
3. The dot product of two unit vectors separated by an angle of 60 degrees would be the cosine of 60 = 1/2.
4. i · j = 0. i · i = 1.
5. (0,0,0) is orthogonal to (1,2,3), and so is (1,1,-1). In general, any vector (x,y,z) with x + 2y + 3z = 0 will be orthogonal to (1,2,3).
6. Projection is additive because the dot product is additive; i.e., proj_v ( w₁ + w₂ ) = = = ( proj_v w₁ ) + ( proj_v w₂ ).
7. The angle between (0,4,-6) and (3,0,-2) is cos^-1 (6/13).
8. The projection of (1,2,3) onto (4,4,4) is ( 8 sqrt(3), 8 sqrt(3), 8 sqrt(3) ).
9. The work done against the force (0,0,-30) by moving an object from (1,1,1) to (-10,3,7) is (0,0,-30) · (-11,2,6) = -180.
10. The Cauchy-Bernard-Schwartz inequality follows from the fact that cosine is always less than 1.
11. The triangle inequality can be geometrically interpreted to mean that the length of one side of a triangle is always less than the sum of the lengths of the other two sides.