Stewart Section 12.3: The Dot Product

1 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2

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

3 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2

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

4 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2

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

5 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

7 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2

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

8 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

12 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

13 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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)

Enter your answer directly below each part above.

25 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

28 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

29 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

30 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

31 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

39 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

56 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

57 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

60 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

62 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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)

Enter your answer directly below each part above.

63 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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)

Enter your answer directly below each part above.

64 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 $

Quick Answer Input

Submit your exam?