3.2 Post-audio exercises
Let u and v be the position vectors (6, 8) and (−12, 5), respectively.
(a) Sketch u and v on a single diagram. On the same diagram, sketch the projection of u onto v, and the projection of v onto u.
(b) Determine the angle between u and v.
(c) Determine the projection of u onto v, and the projection of v onto u.
(b) Let θ denote the angle between u and v. Now
(c) The projection of u onto v is
The projection of v onto u is
Determine the angle between the vectors u = (3, 4, 5) and v = (1, 0, −1) in 3.
Let θ denote the angle between u and v. Now
Find a vector of length 2 that is perpendicular to both of the vectors a = (2, 1, 0) and b = (1, 0, −1).
Let the vector we want be denoted by
v = (x,y,z),
for some real numbers x, y and z.
Since ‖v‖ = 2,
Since v is perpendicular to a,
2x + y = 0,
Since v is perpendicular to b,
x − z = 0,
Substituting the expressions for y in equation (S.4) and for z in equation (S.5) into equation (S.3), we obtain
x2 + (−2x)2 + x2 = 4;
6x2 = 4,
It follows that the two possible vectors that satisfy the given conditions are