Homework Set 3
Section 10.3: The Dot Product
1. Which of the following expressions are meaningful? Which are meaningless? (ie: which can be
computed and which make no sense mathematically?)
(a) (a · b) · c
(b) (a · b) c
(c) |a| (b · c)
(d) a · (b + c)
(e) a · b + c
(f) |a| · (b + c)
2. Determine whether the given vectors are orthogonal, parallel, or neither. If the given pair of
vectors is neither orthogonal nor parallel, find the angle between them.
(a) a = h3, −2i, b = h4, 6i
(b) a = h4, 14, −6i, b = h6, 21, −9i
(c) a = i − j + 2k, b = 2i − j + k
(d) a = 3i − 2j + k, b = j − 5k
Math 2241
hw set 3, page 2 of 3
due January 27
3. Find the value(s) of x such that the angle between u = h2, 1, −1i and v = h1, x, 0i is 45◦ .
4. The vector ortha b = b − proja b is called the orthogonal projection of b.
(a) Show that ortha b is orthogonal to a.
(b) Let a = h−5, 12i and b = h3, 4i for the remainder of the question. Find the scalar and
vector projections of b onto a.
(c) Find ortha b and illustrate the relationships of the vectors a, b, ortha b, and proja b by
sketching them on the xy-plane.
Math 2241
hw set 3, page 3 of 3
due January 27
5. Find the work done by a force 3i + 5j + 10k that moves an object from the point (1, 0, 2) to
the point (5, 3, 8) along a straight line. The distance is measured in meters and the force in
Newtons.
6. A tow truck drags a stalled car along a road. The chain makes an angle of 30◦ with the road,
and the tension in the chain is 1500 N. How much work is done by the truck in pulling the car
1 km?
7. Show that if u + v and u − v are orthogonal, then the vectors u and v must have the same
length. [Hint: this is a proof question, meaning the point of the question is to show all of the
middle steps of going from the orthogonal vectors to the vectors of the same length.]