Assignment 2

Dot Product

1. Find the angle between the vectors ?√

3

1

?

and ?

−

√

3

1

?

.

2. The Triangle Inequality states that any two vectors ~x and ~y in R

n

satisfy

k~x + ~yk ≤ k~xk + k~yk. (1)

(a) Show that ~x =

?

1

−2

?

and ~y =

?

2

1

?

satisfy the Triangle Inequality.

(b) Show that the left and right sides of the inequality are the same for ~x =

?

1

−2

?

and ~y =

?

3

−6

?

. Why is this? (Hint: Sketch the vectors and think about what the inequality is saying

geometrically.)

3. Alex pushes a car with a constant force of 100 N at an angle of

30◦

(or π/6 rad) to the horizontal. In doing so, Alex moves the

car a distance of 10 m. The work (energy) that Alex does is given

by the dot product of the force and displacement, W = F~ ·

~d.

How much work does Alex do in total? (Note: Work is measured

in units of Joules and 1 J is equal to 1 N·m.)

Projections

4. Elena pulls a box with force F~ at an angle θ relative

to the horizontal. Compute the projection and perpendicular of F~ with respect to the horizontal. What

do these represent physically?

5. Consider the line defined by y = −3/2x + 2 and the point P = (5, 5).

(a) Find a vector equation representing the line. Do vectors satisfying this equation form a subspace

of R

2

? Why/Why not?

(b) Q = (4, −4) and R = (0, 2) are two points falling on the line. Let ~x be the vector joining Q

to P and ~y be the vector joining Q to R. Find all of ||~x||, proj~y~x, and perp~y~x. Is one of the

projections the same as before?

(c) Let ~x be the vector joining R to P and ~y be the vector joining Q to R. Find all of ||~x||, proj~y~x,

and perp~y~x.

(d) Based on the above, which vector could we use to describe the minimum distance between P

and a point on the line?

(e) Find the point on the line closest to P.

1

Cross Product

6. Determine the scalar equation of the plane that contains the points P = (1, 5, 3), Q = (2, 6, −1),

and R(1, 0, 1).

7. Let ~x =

1

2

2

and ~y =

−2

−1

1

.

(a) Compute ~w = ~x × ~y. Using the definition of length, find || ~w||.

(b) Compute the product of the following two scalars: ||~x|| and kperp~x~yk. You should get the same

answer as part (a). Why?

(c) Given new vectors ~a =

1

5

5

,

~b (components unknown) and the information that the angle

between ~a and ~b is θ = π/4, find ||proj~b

~a||. Hint: You may find the diagram on p.53 helpful.

8. Suppose ~w = ~u × ~v, ~u, ~v ∈ R

3

.

(a) If ~u · ~v = 0, must ~w = ~0? Under what circumstances will ~w = ~0?

(b) Find a vector ~r orthogonal to ~u, ~v, ~w as defined above when ~u, ~v, ~w 6= ~0. (Hint: No calculation

required.)

Complex Numbers

9. Simplify the following in standard form:

(a) (1 + i

√

3)(1 + i), (b) π + iπ

1 −

√

3i

10. Given the provided root, find the remaining roots of the polynomial.

(a) y = x

3 − 2x

2 + 4x − 3, x = 1

(b) y = x

3 − 4x

2 + 2x + 4, x = 2

2

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