Description
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
