Vectors and conics

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# 3.2 Post-audio exercises

## Example 42

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.

• (a)

• (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

## Example 43

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

and

Thus

## Example 44

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,

thus

2x + y = 0,

that is,

Since v is perpendicular to b,

thus

x − z = 0,

that is,

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;

thus

6x2 = 4,

so

It follows that the two possible vectors that satisfy the given conditions are

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