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Dot Product & Projections
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 11.3 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource)
and your lecture notes.
EXPECTED SKILLS:
• Know how to compute the dot product of two vectors.
• Be able to use the dot product to find the angle between two vectors; and, in particular,
be able to determine if two vectors are orthogonal.
• Know how to compute the direction cosines of a vector.
• Be able to decompose vectors into orthogonal components. And, know how to compute
the orthogonal projection of one vector onto another.
PRACTICE PROBLEMS:
−
−
1. For each of the following, compute →
u ·→
v based on the given information.
−
−
(a) →
u = h3, −1i; →
v = h2, −5i
11
→
−
−
(b) u = h4, −5, 1i; →
v = h3, 6, −1i
−19
−
−
(c) →
u = i − 3j + 4k; →
v = 9i − 2j;
15
π
−
−
−
−
(d) k→
u k = 3; k→
v k = 4; the angle between →
u and →
v is
4
√
6 2
2. Explain why the operation (u · v) · w does not make sense.
u · v is a scalar. We cannot take the dot product of a scalar with a vector.
−
−
3. Determine whether the angle between →
v = h1, 2, 3i and →
w = h−6, 4, −1i is acute,
obtuse, or right. Explain.
−
−
Since →
v ·→
w = −1 < 0, the angle between the two vectors is obtuse.
−
−
4. Give an example of a vector which is orthogonal to both →
v = h1, 1, 1i and →
w = h2, 0, 4i.
(Hint: Set up a system of equations.)
−
Any scalar multiple of →
n = h−2, 1, 1i is orthogonal to both given vectors.
1
5. Consider the triangle with vertices a, b, and c.
Use vectors to compute the angle between the diagonal which extends from vertex a
to vertex b and the line segment which extends from vertex a to vertex c. (Verify your
answer with HW 11.1 #3c.)
√ !
13
cos−1 √
14
6. Consider the triangle, shown below,with vertices A(1, −2, 6), B(3, 0, −1), and C(−2, 1, 0).
Compute all three angles of the triangle.
2
42
√ √
Angle A has a measure of cos
radians.
57 54 15
Angle B has a measure of cos−1 √ √
radians.
57 27 12
Angle C has a measure of cos−1 √ √
radians.
54 27
(NOTE: Once you find any two of the angles, you can use the fact that the sum of all
of the angles must be π radians to compute the remaining angle.)
−1
→
−
−
7. Let →
v = h1, 2i and b = h−3, 4i.
→
−
−
−
(a) Find the vector component of →
v along b and the vector component of →
v or→
−
thogonal to b .
→
−
3 4
→
−
→
−
→ v =
The vector component of v along b is Proj−
and the vector
− ,
b
55
→
−
8 6
−
−
→
−
→ v =
,
component of →
v orthogonal to b is →
v − Proj−
.
b
5 5
→
−
−
(b) Sketch →
v , b , and the vector components that you found in part (a).
8. Express v = i + 2j + 3k as the sum of a vector parallel to b = −2i + 4j − k and a
vector perpendicular to b
2 4 1
9 10 22
v = − , ,−
+
, ,
; Detailed Solution: Here
7 7 7
7 7 7
9. Suppose that u 6= 0 and v 6= 0. Under what condition will ku + vk = ku − vk?
Explain.
The result follows if u is orthogonal to v; Detailed Solution: Here
3
10. The following questions deal with finding the distance from a point to a line:
(a) Given three points A, B, and P in 2-space or 3-space as shown in the picture
below, describe two different ways that you could use the dot product to calculate
the distance, d, between the point P and the line which contains A and B.
OPTION 1:
−→
−→
→ AP . Then, use Pythagorean Theorem to
One can compute AP and Proj−
AB
calculate d.
OPTION 2:
−→
−→
−→
−
→
−
→
One can compute ProjAB AP . Then, d = AP − ProjAB AP .
(b) Use one of your methods from part (a) to compute the distance from the point
P (5, 3, 0) to the line containing A(1, 0, 1) and B(2, 3, 1).
r
91
; Video Solution: Here
d=
10
11. Consider the triangle shown below which is formed by vectors v and w.
Prove Pythagorean’s Theorem kv + wk2 = kvk2 + kwk2 . (Hint: Use properties of the
dot product to expand kv + wk2 .)
4
We expand kv + wk2 using properties of the dot product:
kv + wk2 = (v + w) · (v + w)
=v·v+v·w+w·v+w·w
= kvk2 + 2(v · w) + kwk2 (since v · w = w · v)
= kvk2 + kwk2 (since v ⊥ w ⇔ v · w = 0)
Thus, kv + wk2 = kvk2 + kwk2 , as promised.
12. Let A and B be endpoints of a diameter of a circle with a radius of r . And, suppose
that C is any other point on the circle, as shown below.
−→
−−→
Prove that triangle ABC is a right triangle. (Hint: Express each of CA and CB as
−→
the combination of CO and some other vector.)
Notice that CA = CO + OA and CB = CO + OB. But, OB = −OA. Thus,
CB = CO − OA. We will show that CA ⊥ CB by showing that CA · CB = 0.
CA · CB = (CO + OA) · (CO − OA)
= CO · CO − CO · OA + OA · CO − OA · OA
= kCOk2 − kOAk2
= r2 − r2
=0
So, CA ⊥ CB and the triangle is a right triangle.
5
−
−
13. Let →
v and →
w be vectors, either both in R2 or in R3 . Prove the Cauchy-Schwarz
−
−
−
−
Inequality: |→
v ·→
w | ≤ k→
v kk→
w k.
−
−
Suppose that θ is the angle between →
v and →
w . Then:
−
−
−
−
|→
v ·→
w | = |k→
v kk→
w k cos θ|
→
−
−
= (k v k) (k→
w k) (| cos θ|)
→
−
→
−
≤ (k v k) (k w k) (1)
−
−
= k→
v kk→
wk
−
−
−
−
Thus, |→
v ·→
w | ≤ k→
v kk→
w k, as promised.
−
14. Let →
v = h1, 2, 3i.
−
(a) Compute the direction cosines of →
v.
−
Let α, β, and γ be the angles between →
v and i, j, and k, respectively. Then:
1
2
3
cos α = √ , cos β = √ , and cos γ = √
14
14
14
(b) Verify that cos2 α + cos2 β + cos2 γ = 1, where α, β, and γ be the angles between
→
−
v and i, j, and k, respectively.
Using the information from part (a), we obtain:
2
2
2
cos α + cos β + cos γ =
1
√
14
2
+
1
4
9
+
+
14 14 14
=1
=
6
2
√
14
2
+
3
√
14
2