7.2 Dot Product
KEY CONCEPTS
r r
r r
∣a ∣∣b ∣cos θ, where θ is the
• The dot product is defined
as
a
⋅
b
=
r
r
angle between a and b , 0º ≤ θ ≤ 180º.
bជ
r
r r
r
• For any vectors u , v , and w and scalar k ∊ ,
θ
r
r
r r
• u ≠ 0 and v ≠ 0 are perpendicular if and only if u ⋅ v = 0
aជ
r r r r
• u ⋅ v = v ⋅ u (commutative property)
r r
r r
r
r
• (k u ) ⋅ v = k (u ⋅ v ) = u ⋅ (kv ) (associative property)
r r
r
r r r r
r
r r r r
r r
• u ⋅ (v + w) = u ⋅ v + u ⋅ w and (v + w) ⋅ u = v ⋅ u + w ⋅ u (distributive property)
r r
r
• u ⋅ u = ∣u∣2
r r
• u ⋅0 = 0
r
r
r r
• If u = [u1, u2] and v = [v1, v2], then u ⋅ v = u1v1 + u2v2.
A
1. Calculate the dot product for each pair
of vectors. Round answers to one decimal
place.
a)
a ⫽ 125
60°
b ⫽ 178
b)
u ⫽ 80
110°
v ⫽ 104
r ⫽ 1150
c)
s ⫽ 1300
d)
2. Calculate the dot product for each pair
of vectors. The angle between the two
vectors when placed tail to tail is θ.
Round answers to one decimal place.
r
r
a) ∣a∣ = 5, ∣b∣ = 9, θ = 80°
r
r
b) ∣m∣ = 34, ∣n∣ = 53, θ = 115°
r
r
c) ∣u∣ = 7.1, ∣v ∣ = 11.7, θ = 270°
r
r
d) ∣r ∣ = 1250, ∣s ∣ = 1620, θ = 240°
r
r
e) ∣v ∣ = 15.6, ∣ w∣ = 23.8, θ = 190°
r
r
f) ∣ p∣ = 66, ∣q∣ = 92, θ = 83°
3. Use the properties of the dot product
to expand and simplify each of the
following, where k, ℓ ∊ .
r r
r
a) (a − k b ) ⋅ c
r r
r
b) k u ⋅ (v + ℓw)
r
r
r r
c) (a + kb ) ⋅ (a − b )
r
r
r r
d) (k u − w) ⋅ (a + ℓb )
n ⫽ 625
m
⫽ 570
180°
7.2 Dot Product • MHR
123
4. Calculate the dot product of each pair
of vectors.
r
r
a) a = [−1, 5], b = [3, 2]
r
r
b) m = [0, −4], n = [6, −5]
r
r
c) p = [3, 8], q = [−9, −7]
r
r
d) u = [4, 0], v = [−1, −1]
r
r
e) r = [7, −3], s = [5, −5]
r
r
f) c = [−6, 2], d = [11, −9]
5. State whether each expression has any
meaning, where k∊. If not, explain
why not.
rr
r r r
b) v ⋅ (w + u )
a) k ⋅ (bc )
r r
r
d) a 3
c) ∣u ⋅ u∣
r
r
f) k + m
e) ∣a∣3
B
6. The angle between two vectors is θ.
a) For what values of θ is the dot
product of the vectors positive?
Explain.
b) For what values of θ is the dot
product of the vectors negative?
Explain.
r
r
7. Let u = [4, −4], v = [7, −2], and
r
w = [−5, 3]. Evaluate each of the
following, if possible. If not possible,
explain why not.
r r r
a) (w − u ) ⋅ v
rr
b) v u
r r r r
c) u ⋅ w + u ⋅ u
r r
r
d) (2 u ) ⋅ (v − 4 w)
r r r
e) w ⋅ v + u
r
r
r r
f) (7v ) ⋅ (−3u ) + w ⋅ w
r r r
g) w ⋅ v ⋅ u
r r r r
h) (w + v )(v − u )
r r
r r
i) (w + v ) ⋅ (v − u )
r r
j) (−w ) ⋅ u
124
8. a) Determine three vectors that are
r
perpendicular to m = [−3, 4].
b) Determine a unit vector that is
perpendicular to the vector in part a).
Is this answer unique? Explain.
9. Simplifyreach rexpression for the unit
vectors i and j .
r r
a) j ⋅ j
r r
b) −i ⋅ j
r
r r
c) (−2 i + 7 j ) ⋅ j
r r r r
d) ( j ⋅ 4 i )(i −3 j )
10. Use Cartesian vectors to prove the results
in question 9.
r
r
r
r
r
r
11. Given a = 3 i + m j and b = −4 i + 2 j ,
determine the value of m in each case.
r
r
a) a and b are perpendicular.
r
r
b) a and b are parallel.
r
r
c) The angle between a and b , when
placed tail to tail, is 60°.
12. Determine if each triangle is a right
triangle. If so, identify the right angle.
a) △ABC with vertices A(5, 6), B(0, 2),
and C(−4, 3)
b) △PQR with vertices P(−5, 3),
Q(−7, 8), and R(3, 12)
13. a) Consider a triangle with vertices
A(−2, 3), B(0, 0), and C(3, 2). Show
that △ABC is a right triangle.
b) Determine the coordinates of a point
D such that ABCD forms a rectangle.
Use the dot product to verify your
answer.
MHR • Chapter 7 Cartesian Vectors 978-0-07-073589-7
14. Determine the value of k, k ∊ , so that
the vectors in each pair are perpendicular.
r
r
a) a = [k, 5], b = [2, −4]
r
r
b) m = [7, k], n = [3, −1]
r
r
c) p = [−1, 2], q = [2k, −3]
r
r
d) u = [3k, −k], v = [5, −5]
r
r
e) r = [−4k, 3], s = [1, k]
r
r
f ) c = [−6, 8k], d = [3, −3]
15. Use an example to verify each statement
r
r r
for vectors u , v , and w.
r r
r
r r r r
a) u ⋅ (v − w) = u ⋅ v − u ⋅ w
r r r
r r r r
b) (u − v ) ⋅ w = u ⋅ w − v ⋅ w
16. Use a counterexample to prove that
the following statement is false.
r r r r
r r
If u ⋅ u = v ⋅ v , then u = v .
r r
r
r
r
r
17. Given a = −2 i +r 5 j and b =r i − 7 j ,
r
r
determine (4 a − b ) ⋅ (3 a + 2b ).
r
r
18. a) If vectors u and v are perpendicular,
r r2
r
r
prove that ∣u + v ∣ = ∣u∣2 + ∣v ∣2.
b) What important result does the
relationship in part a) represent?
r
r
19. Let a = (a1, a2) and b = (b1, b2).
r
r
a) Express a and b in terms of unit
vectors.
b) Use your expressions
from part a) to
r r
prove that a ⋅ b = a1b1 + a2b2.
20. Determine the coordinates of a vector
r
r
a such that a ⋅ (−3, 2) = 1 and
r
(1, 4) ⋅ a = 5.
r r r r
21. Suppose (a −rb ) ⋅ (a r+ b ) = 0. What
is
r
r
r r
true about a , b , a + b , and a − b ? Justify
your answer geometrically.
r
r
22. Consider the non-zero vectors u and v .
r
r
Show that u and v are perpendicular if
r r
r r
∣u + v ∣ = ∣u − v ∣.
r r
r r
r r
23. Prove that 4 a ⋅ b =r ∣a + b∣2 − ∣a − b∣2 for
r
any vectors a and b .
C
r
r
r r
24. Determine (3
a − 4 b ) ⋅ (2 a + b ) given
r
r
that a and b are unit vectors and the
angle between them is 30°.
r
r
r
r
25. Determine (5 a − 2 b ) ⋅ (3 a + 8 b ) given
r
r
that ∣a∣ = 4, ∣b∣ =
r 9, and the angle
r
between a and b is 120°.
r
r
26. Suppose ar and b__ are unit vectors such
r
∣ √
that ∣a +
r b =r 7 . Determine
r
r
(a + 5 b ) ⋅ (3 a − 2 b ).
r r
r
27. Three vectors u , v ,rand w exist such
r r
r
that u + v − w = 0 . Determine the value
r r r r
r r
of u ⋅ v +
r v ⋅ w + w ⋅ru given that
r
∣u∣= 4, ∣v ∣ = 3, and ∣w∣ = 6.
r
r r r
r r
28. a) For any vectors u and v , ∣u ⋅ v ∣ ≤ ∣u∣∣v ∣.
Prove this statement is true.
b) Under what condition does the
equality hold true?
r r
r
29. If u , v , and w are non-zero vectors such
r r
r
r r r
that
), prove that
r u ⋅ (v + w) = v ⋅ (ur − w
r
w is perpendicular to (u + v ).
r r
r
30. If u , v , andrw rarer non-zero
r r vectors
r
such
r that
r (u ⋅ v )w = (v ⋅ w) u , prove that
w and u are parallel.
31. Use the dot product to show that if
the diagonals of a parallelogram are
perpendicular, then the parallelogram is a
rhombus.
7.2 Dot Product • MHR
125