Stewart Section 12.3: The Dot Product

1 Vectors - Dot Product · Level 2

Which of the following expressions are meaningful? Which are meaningless? Explain.

(a) $(\mathbf{a} \cdot \mathbf{b}) \cdot \mathbf{c}$

Answer for 1(a)

(b) $(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$

Answer for 1(b)

(c) $|\mathbf{a}|(\mathbf{b} \cdot \mathbf{c})$

Answer for 1(c)

(d) $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})$

Answer for 1(d)

(e) $\mathbf{a} \cdot \mathbf{b} + \mathbf{c}$

Answer for 1(e)

(f) $|\mathbf{a}| \cdot (\mathbf{b} + \mathbf{c})$

Answer for 1(f)

Enter your answer directly below each part above.

2 Vectors - Dot Product · Level 2

Find $\mathbf{a} \cdot \mathbf{b}$. $\mathbf{a} = \langle 5, -2 \rangle$, $\mathbf{b} = \langle 3, 4 \rangle$

3 Vectors - Dot Product · Level 2

Find $\mathbf{a} \cdot \mathbf{b}$. $\mathbf{a} = \langle 1.5, 0.4 \rangle$, $\mathbf{b} = \langle -4, 6 \rangle$

4 Vectors - Dot Product · Level 2

Find $\mathbf{a} \cdot \mathbf{b}$. $\mathbf{a} = \langle 6, -2, 3 \rangle$, $\mathbf{b} = \langle 2, 5, -1 \rangle$

5 Vectors - Dot Product · Level 2

Find $\mathbf{a} \cdot \mathbf{b}$. $\mathbf{a} = \langle 4, 1, \dfrac{1}{4} \rangle$, $\mathbf{b} = \langle 6, -3, -8 \rangle$

6 Vectors - Dot Product · Level 3

Find $\mathbf{a} \cdot \mathbf{b}$. $\mathbf{a} = \langle p, -p, 2p \rangle$, $\mathbf{b} = \langle 2q, q, -q \rangle$

7 Vectors - Dot Product · Level 2

Find $\mathbf{a} \cdot \mathbf{b}$. $\mathbf{a} = 2 \mathbf{i} + \mathbf{j}$, $\mathbf{b} = \mathbf{i} - \mathbf{j} + \mathbf{k}$

8 Vectors - Dot Product · Level 2

Find $\mathbf{a} \cdot \mathbf{b}$. $\mathbf{a} = 3 \mathbf{i} + 2 \mathbf{j} - \mathbf{k}$, $\mathbf{b} = 4 \mathbf{i} + 5 \mathbf{k}$

9 Vectors - Dot Product · Level 2

Find $\mathbf{a} \cdot \mathbf{b}$. $|\mathbf{a}| = 7$, $|\mathbf{b}| = 4$, the angle between $\mathbf{a}$ and $\mathbf{b}$ is $30^{\circ}$

10 Vectors - Dot Product · Level 2

Find $\mathbf{a} \cdot \mathbf{b}$. $|\mathbf{a}| = 80$, $|\mathbf{b}| = 50$, the angle between $\mathbf{a}$ and $\mathbf{b}$ is $3 \dfrac{\pi}{4}$

11 Vectors - Dot Product · Level 3

If $\mathbf{u}$ is a unit vector, find $\mathbf{u} \cdot \mathbf{v}$ and $\mathbf{u} \cdot \mathbf{w}$.

12 Vectors - Dot Product · Level 3

If $\mathbf{u}$ is a unit vector, find $\mathbf{u} \cdot \mathbf{v}$ and $\mathbf{u} \cdot \mathbf{w}$.

13 Vectors - Dot Product Properties · Level 3

(a) Show that $\mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} = 0$.

Answer for 13(a)

(b) Show that $\mathbf{i} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{j} = \mathbf{k} \cdot \mathbf{k} = 1$.

Answer for 13(b)

Enter your answer directly below each part above.

14 Vectors - Dot Product Applications · Level 2

A street vendor sells $a$ hamburgers, $b$ hot dogs, and $c$ soft drinks on a given day. He charges \$4 for a hamburger, \$2.50 for a hot dog, and \$1 for a soft drink. If $\mathbf{A} = \langle a, b, c \rangle$ and $\mathbf{P} = \langle 4, 2.5, 1 \rangle$, what is the meaning of the dot product $\mathbf{A} \cdot \mathbf{P}$?

15 Vectors - Angle Between Vectors · Level 3

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $\mathbf{a} = \langle 4, 3 \rangle$, $\mathbf{b} = \langle 2, -1 \rangle$

16 Vectors - Angle Between Vectors · Level 3

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $\mathbf{a} = \langle -2, 5 \rangle$, $\mathbf{b} = \langle 5, 12 \rangle$

17 Vectors - Angle Between Vectors · Level 3

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $\mathbf{a} = \langle 1, -4, 1 \rangle$, $\mathbf{b} = \langle 0, 2, -2 \rangle$

18 Vectors - Angle Between Vectors · Level 3

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $\mathbf{a} = \langle -1, 3, 4 \rangle$, $\mathbf{b} = \langle 5, 2, 1 \rangle$

19 Vectors - Angle Between Vectors · Level 3

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $\mathbf{a} = 4 \mathbf{i} - 3 \mathbf{j} + \mathbf{k}$, $\mathbf{b} = 2 \mathbf{i} - \mathbf{k}$

20 Vectors - Angle Between Vectors · Level 3

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $\mathbf{a} = 8 \mathbf{i} - \mathbf{j} + 4 \mathbf{k}$, $\mathbf{b} = 4 \mathbf{j} + 2 \mathbf{k}$

21 Vectors - Angle Between Vectors · Level 3

Find, correct to the nearest degree, the three angles of the triangle with the given vertices. $P(2, 0)$, $Q(0, 3)$, $R(3, 4)$

22 Vectors - Angle Between Vectors · Level 3

Find, correct to the nearest degree, the three angles of the triangle with the given vertices. $A(1, 0, -1)$, $B(3, -2, 0)$, $C(1, 3, 3)$

23 Vectors - Orthogonality · Level 3

Determine whether the given vectors are orthogonal, parallel, or neither.

(a) $\mathbf{a} = \langle 9, 3 \rangle$, $\mathbf{b} = \langle -2, 6 \rangle$

Answer for 23(a)

(b) $\mathbf{a} = \langle 4, 5, -2 \rangle$, $\mathbf{b} = \langle 3, -1, 5 \rangle$

Answer for 23(b)

(c) $\mathbf{a} = -8 \mathbf{i} + 12 \mathbf{j} + 4 \mathbf{k}$, $\mathbf{b} = 6 \mathbf{i} - 9 \mathbf{j} - 3 \mathbf{k}$

Answer for 23(c)

(d) $\mathbf{a} = 3 \mathbf{i} - \mathbf{j} + 3 \mathbf{k}$, $\mathbf{b} = 5 \mathbf{i} + 9 \mathbf{j} - 2 \mathbf{k}$

Answer for 23(d)

Enter your answer directly below each part above.

24 Vectors - Orthogonality · Level 3

Determine whether the given vectors are orthogonal, parallel, or neither.

(a) $\mathbf{u} = \langle -5, 4, -2 \rangle$, $\mathbf{v} = \langle 3, 4, -1 \rangle$

Answer for 24(a)

(b) $\mathbf{u} = 9 \mathbf{i} - 6 \mathbf{j} + 3 \mathbf{k}$, $\mathbf{v} = -6 \mathbf{i} + 4 \mathbf{j} - 2 \mathbf{k}$

Answer for 24(b)

(c) $\mathbf{u} = \langle c, c, c \rangle$, $\mathbf{v} = \langle c, 0, -c \rangle$

Answer for 24(c)

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25 Vectors - Orthogonality · Level 3

Use vectors to decide whether the triangle with vertices $P(1, 3, -2)$, $Q(2, 0, -4)$, and $R(6, -2, -5)$ is right-angled.

26 Vectors - Angle Between Vectors · Level 3

Find the values of $x$ such that the angle between the vectors $\langle 2, 1, -1 \rangle$ and $\langle 1, x, 0 \rangle$ is $45^{\circ}$.

27 Vectors - Orthogonality · Level 3

Find a unit vector that is orthogonal to both $\mathbf{i} + \mathbf{j}$ and $\mathbf{i} + \mathbf{k}$.

28 Vectors - Angle Between Vectors · Level 4

Find two unit vectors that make an angle of $60^{\circ}$ with $\mathbf{v} = \langle 3, 4 \rangle$.

29 Vectors - Angle Between Lines · Level 3

Find the acute angle between the lines. $2x - y = 3$, $3x + y = 7$

30 Vectors - Angle Between Lines · Level 3

Find the acute angle between the lines. $x + 2y = 7$, $5x - y = 2$

31 Vectors - Angle Between Curves · Level 4

Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) $y = x^2$, $y = x^3$

32 Vectors - Angle Between Curves · Level 4

Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) $y = \sin x$, $y = \cos x$, $0 \leq x \leq \dfrac{\pi}{2}$

33 Vectors - Direction Cosines · Level 3

Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) $\langle 2, 1, 2 \rangle$

34 Vectors - Direction Cosines · Level 3

Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) $\langle 6, 3, -2 \rangle$

35 Vectors - Direction Cosines · Level 3

Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) $\mathbf{i} - 2 \mathbf{j} - 3 \mathbf{k}$

36 Vectors - Direction Cosines · Level 3

Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) $\dfrac{1}{2} \mathbf{i} + \mathbf{j} + \mathbf{k}$

37 Vectors - Direction Cosines · Level 3

Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) $\langle c, c, c \rangle$, where $c > 0$

38 Vectors - Direction Cosines · Level 4

If a vector has direction angles $\alpha = \dfrac{\pi}{4}$ and $\beta = \dfrac{\pi}{3}$, find the third direction angle $\gamma$.

39 Vectors - Projections · Level 3

Find the scalar and vector projections of $\mathbf{b}$ onto $\mathbf{a}$. $\mathbf{a} = \langle -5, 12 \rangle$, $\mathbf{b} = \langle 4, 6 \rangle$

40 Vectors - Projections · Level 3

Find the scalar and vector projections of $\mathbf{b}$ onto $\mathbf{a}$. $\mathbf{a} = \langle 1, 4 \rangle$, $\mathbf{b} = \langle 2, 3 \rangle$

41 Vectors - Projections · Level 3

Find the scalar and vector projections of $\mathbf{b}$ onto $\mathbf{a}$. $\mathbf{a} = \langle 4, 7, -4 \rangle$, $\mathbf{b} = \langle 3, -1, 1 \rangle$

42 Vectors - Projections · Level 3

Find the scalar and vector projections of $\mathbf{b}$ onto $\mathbf{a}$. $\mathbf{a} = \langle -1, 4, 8 \rangle$, $\mathbf{b} = \langle 12, 1, 2 \rangle$

43 Vectors - Projections · Level 3

Find the scalar and vector projections of $\mathbf{b}$ onto $\mathbf{a}$. $\mathbf{a} = 3 \mathbf{i} - 3 \mathbf{j} + \mathbf{k}$, $\mathbf{b} = 2 \mathbf{i} + 4 \mathbf{j} - \mathbf{k}$

44 Vectors - Projections · Level 3

Find the scalar and vector projections of $\mathbf{b}$ onto $\mathbf{a}$. $\mathbf{a} = \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}$, $\mathbf{b} = 5 \mathbf{i} - \mathbf{k}$

45 Vectors - Projections · Level 4

Show that the vector $\text{orth}_\mathbf{a} \mathbf{b} = \mathbf{b} - \text{proj}_\mathbf{a} \mathbf{b}$ is orthogonal to $\mathbf{a}$. (It is called an orthogonal projection of $\mathbf{b}$.)

46 Vectors - Projections · Level 3

For the vectors in Exercise 40, find $\text{orth}_\mathbf{a} \mathbf{b}$ and illustrate by drawing the vectors $\mathbf{a}$, $\mathbf{b}$, $\text{proj}_\mathbf{a} \mathbf{b}$, and $\text{orth}_\mathbf{a} \mathbf{b}$.

47 Vectors - Projections · Level 3

If $\mathbf{a} = \langle 3, 0, -1 \rangle$, find a vector $\mathbf{b}$ such that $\text{comp}_\mathbf{a} \mathbf{b} = 2$.

48 Vectors - Projections · Level 4

Suppose that $\mathbf{a}$ and $\mathbf{b}$ are nonzero vectors.

(a) Under what circumstances is $\text{comp}_\mathbf{a} \mathbf{b} = \text{comp}_\mathbf{b} \mathbf{a}$?

Answer for 48(a)

(b) Under what circumstances is $\text{proj}_\mathbf{a} \mathbf{b} = \text{proj}_\mathbf{b} \mathbf{a}$?

Answer for 48(b)

Enter your answer directly below each part above.

49 Vectors - Work · Level 3

Find the work done by a force $\mathbf{F} = 8 \mathbf{i} - 6 \mathbf{j} + 9 \mathbf{k}$ that moves an object from the point $(0, 10, 8)$ to the point $(6, 12, 20)$ along a straight line. The distance is measured in meters and the force in newtons.

50 Vectors - Work · Level 3

A tow truck drags a stalled car along a road. The chain makes an angle of $30^{\circ}$ 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?

51 Vectors - Work · Level 3

A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\circ}$ above the horizontal moves the sled 80 ft. Find the work done by the force.

52 Vectors - Work · Level 3

A boat sails south with the help of a wind blowing in the direction S36${^{\circ}}$E with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft.

53 Vectors - Projections · Level 4

Use a scalar projection to show that the distance from a point $P_1 (x_1, y_1)$ to the line $a x + b y + c = 0$ is $ \dfrac{|a x_1 + b y_1 + c|}{\sqrt{a^2 + b^2}} $ Use this formula to find the distance from the point $(-2, 3)$ to the line $3x - 4y + 5 = 0$.

54 Vectors - Dot Product Applications · Level 4

If $\mathbf{r} = \langle x, y, z \rangle$, $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$, and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$, show that the vector equation $(\mathbf{r} - \mathbf{a}) \cdot (\mathbf{r} - \mathbf{b}) = 0$ represents a sphere, and find its center and radius.

55 Vectors - Angle Between Vectors · Level 3

Find the angle between a diagonal of a cube and one of its edges.

56 Vectors - Angle Between Vectors · Level 3

Find the angle between a diagonal of a cube and a diagonal of one of its faces.

57 Vectors - Dot Product Applications · Level 4

A molecule of methane, $\text{CH}_4$, is structured with the four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the H--C--H combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about $109.5^{\circ}$. [Hint: Take the vertices of the tetrahedron to be the points $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$, and $(1, 1, 1)$, as shown in the figure. Then the centroid is $\left(\dfrac{1}{2}, \dfrac{1}{2}, \dfrac{1}{2}\right)$.]

58 Vectors - Dot Product Properties · Level 4

If $\mathbf{c} = |\mathbf{a}| \mathbf{b} + |\mathbf{b}| \mathbf{a}$, where $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are all nonzero vectors, show that $\mathbf{c}$ bisects the angle between $\mathbf{a}$ and $\mathbf{b}$.

59 Vectors - Dot Product Properties · Level 4

Prove Properties 2, 4, and 5 of the dot product (Theorem 2).

60 Vectors - Dot Product Properties · Level 4

Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

61 Vectors - Dot Product Properties · Level 4

Use Theorem 3 to prove the Cauchy-Schwarz Inequality: $ |\mathbf{a} \cdot \mathbf{b}| \leq |\mathbf{a}| |\mathbf{b}| $

62 Vectors - Dot Product Properties · Level 4

The Triangle Inequality for vectors is $ |\mathbf{a} + \mathbf{b}| \leq |\mathbf{a}| + |\mathbf{b}| $

(a) Give a geometric interpretation of the Triangle Inequality.

Answer for 62(a)

(b) Use the Cauchy-Schwarz Inequality from Exercise 61 to prove the Triangle Inequality. [Hint: Use the fact that $|\mathbf{a} + \mathbf{b}|^2 = (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b})$ and use Property 3 of the dot product.]

Answer for 62(b)

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63 Vectors - Dot Product Properties · Level 4

The Parallelogram Law states that $ |\mathbf{a} + \mathbf{b}|^2 + |\mathbf{a} - \mathbf{b}|^2 = 2 |\mathbf{a}|^2 + 2 |\mathbf{b}|^2 $

(a) Give a geometric interpretation of the Parallelogram Law.

Answer for 63(a)

(b) Prove the Parallelogram Law. (See the hint in Exercise 62.)

Answer for 63(b)

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64 Vectors - Dot Product Properties · Level 4

Show that if $\mathbf{u} + \mathbf{v}$ and $\mathbf{u} - \mathbf{v}$ are orthogonal, then the vectors $\mathbf{u}$ and $\mathbf{v}$ must have the same length.

65 Vectors - Projections · Level 5

If $\theta$ is the angle between vectors $\mathbf{a}$ and $\mathbf{b}$, show that $ \text{proj}_\mathbf{a} \mathbf{b} \cdot \text{proj}_\mathbf{b} \mathbf{a} = (\mathbf{a} \cdot \mathbf{b}) \cos^2 \theta $

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