7.5e: The Dot Product

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A: Evaluate the Dot Product

Exercise $\PageIndex{A}$

Given vectors $\vecs u$ and $\vecs v$, calculate the dot product $\vecs u \cdot \vecs v$

1) $\quad \vecs{u}=⟨3,0⟩, \quad \vecs{v}=⟨2,2⟩$

2) $\quad \vecs{u}=⟨3,−4⟩, \quad \vecs{v}=⟨4,3⟩$

3) $\quad \vecs{u}=⟨2,2,−1⟩, \quad \vecs{v}=⟨−1,2,2⟩$

4) $\quad \vecs{u}=⟨4,5,−6⟩, \quad \vecs{v}=⟨0,−2,−3⟩$

5) $\quad \vecs u=\left \langle -1,6 \right \rangle, \quad \vecs v=\left \langle 6,-1 \right \rangle$

6) $\quad \vecs u=\left \langle -2,4 \right \rangle, \quad \vecs v=\left \langle -3,1 \right \rangle$

7) $\quad \vecs u = −\mathbf{\hat i} − \mathbf{\hat j}, \quad \vecs v = \mathbf{\hat i} + 5\mathbf{\hat j}$

8) $\quad \vecs u = 3\mathbf{\hat i} − 4\mathbf{\hat j}, \quad \vecs v = −2\mathbf{\hat i} + 3\mathbf{\hat j}$

Given vectors $\vecs{a}, \,\vecs{b}$, and $\vecs{c}$, determine the vectors $(\vecs{a}\cdot\vecs{b})\vecs{c}$ and $(\vecs{a}⋅\vecs{c})\vecs{b}.$ Express the vectors in component form.

11) $\quad \vecs{a}=⟨2,0,−3⟩, \quad \vecs{b}=⟨−4,−7,1⟩, \quad \vecs{c}=⟨1,1,−1⟩$

12) $\quad \vecs{a}=⟨0,1,2⟩, \quad \vecs{b}=⟨−1,0,1⟩, \quad \vecs{c}=⟨1,0,−1⟩$

13) $\quad \vecs{a}=\mathbf{\hat i} +\mathbf{\hat j} , \quad \vecs{b}=\mathbf{\hat i} −\mathbf{\hat k} , \quad \vecs{c}=\mathbf{\hat i} −2\mathbf{\hat k} $

14) $\quad \vecs{a}=\mathbf{\hat i} −\mathbf{\hat j} +\mathbf{\hat k} , \quad \vecs{b}=\mathbf{\hat j} +3\mathbf{\hat k} , \quad \vecs{c}=−\mathbf{\hat i} +2\mathbf{\hat j} −4\mathbf{\hat k} $

Answers to odd exercises:: 1. $6 \qquad$ 3. $0 \qquad$ 5. $-12 \qquad $ 7. $-6 \qquad$ 11. $ (\vecs a⋅\vecs b)\vecs c=⟨−11,−11,11⟩; \quad (\vecs a⋅\vecs c)b=⟨−20,−35,5⟩$
13. $ (\vecs a⋅\vecs b)\vecs c=⟨1,0,−2⟩; \quad (\vecs a⋅\vecs c)\vecs b=⟨1,0,−1⟩$

B: Find the angle between 2 Vectors

Exercise $\PageIndex{B}$

Given two vectors,

a. Find the measure of the angle $\displaystyle θ$ between a and b. Express the answer in degrees rounded to the nearest tenth, if it is not possible to express it exactly.

b. Is $ θ$ an acute angle?

21) $\quad \vecs{a}=⟨3,−1⟩, \quad \vecs{b}=⟨−4,0⟩$

22) $\quad \vecs{a}=⟨2,1⟩, \quad \vecs{b}=⟨−1,3⟩$

23) $\quad \vecs{u}=3\mathbf{\hat i}, \quad \vecs{v}=4\mathbf{\hat i} +4\mathbf{\hat j} $

24) $\quad \vecs{u}=5\mathbf{\hat i}, \quad \vecs{v}=−6\mathbf{\hat i} +6\mathbf{\hat j} $

Find the measure of the angle between the three-dimensional vectors $\vecs{a}$ and $\vecs{b}$. Express the answer in degrees rounded to the nearest tenth, if it is not possible to express it exactly.

25) $\quad \vecs{a}=⟨3,−1,2⟩, \quad \vecs{b}=⟨1,−1,−2⟩$

26) $\quad \vecs{a}=⟨0,−1,−3⟩, \quad \vecs{b}=⟨2,3,−1⟩$

27) $\quad \vecs{a}=\mathbf{\hat i} +\mathbf{\hat j} , \quad \vecs{b}=\mathbf{\hat j} −\mathbf{\hat k} $

28) $\quad \vecs{a}=\mathbf{\hat i} −2\mathbf{\hat j} +\mathbf{\hat k} , \quad \vecs{b}=\mathbf{\hat i} +\mathbf{\hat j} −2\mathbf{\hat k} $

29) $\quad \vecs{a}=3\mathbf{\hat i} −\mathbf{\hat j} −2\mathbf{\hat k} , \quad \vecs{b}=\vecs v+\vecs w,$ where $\quad \vecs{v}=−2\mathbf{\hat i} −3\mathbf{\hat j} +2\mathbf{\hat k} $ and $\vecs{w}=\mathbf{\hat i} +2\mathbf{\hat k} $

30) $\quad \vecs{a}=3\mathbf{\hat i} −\mathbf{\hat j} +2\mathbf{\hat k} , \quad \vecs{b}=\vecs v−\vecs w,$ where $\quad \vecs{v}=2\mathbf{\hat i} +\mathbf{\hat j} +4\mathbf{\hat k} $ and $\vecs{w}=6\mathbf{\hat i} +\mathbf{\hat j} +2\mathbf{\hat k} $

33) Consider the points $P(4,5)$ and $Q(5,−7)$, and note that $O$ represents the origin.

a. Determine vectors $\vecd{OP}$ and $\vecd{OQ}$. Express the answer by using standard unit vectors.

b. Determine the measure of angle $O$ in triangle $OPQ$. Express the answer in degrees rounded to two decimal places.

34) Consider points $ A(1,1), B(2,−7),$ and $C(6,3)$.

a. Determine vectors $ \vecd{BA}$ and $\vecd{BC}$. Express the answer in component form.

b. Determine the measure of angle $B$ in triangle $ABC$. Express the answer in degrees rounded to two decimal places.

35) Determine the measure of angle $A$ in triangle $ABC$, where $A(1,1,8), B(4,−3,−4),$ and $C(−3,1,5).$ Express your answer in degrees rounded to two decimal places.

36) Consider points $\displaystyle P(3,7,−2)$ and $\displaystyle Q(1,1,−3)$, with $O$ representing the origin. Determine the angle between vectors $\vecd{OP}$ and $\displaystyle \vec{OQ}$. Express the answer in degrees rounded to two decimal places.

$This figure is the 3-dimensional coordinate system. There are four points plotted. The first point is labeled “P(1, 1, -1),” the second point is labeled “Q(1, -1, 1),” the third point is labeled “R(-1, 1, 1),” and the fourth point is labeled “S(-1, -1, -1).” There are line segments from Q to P, P to R and R to P. There are also two vectors in standard position. The first has terminal point of R and the second has terminal point of S. The angle between them is represented with an arc.$ 37. A methane molecule has a carbon atom situated at the origin and four hydrogen atoms located at points $\displaystyle P(1,1,−1),Q(1,−1,1),R(−1,1,1),$ and $\displaystyle S(−1,−1,−1)$ (see figure).

a. Find the distance between the hydrogen atoms located at P and R.

b. Find the angle between vectors $\displaystyle \vec{OS}$ and $\displaystyle \vec{OR}$ that connect the carbon atom with the hydrogen atoms located at S and R, which is also called the bond angle. Express the answer in degrees rounded to two decimal places.

38. Find the vectors that join the center of a clock to the hours 1:00, 2:00, and 3:00. Assume the clock is circular with a radius of 1 unit.

Answers to odd exercises:: 21. $\displaystyle a. θ= 161.6°$; $ \displaystyle b. θ$ is not acute. 23. $\displaystyle a. θ=45°$; $ \; \displaystyle b. θ$ is acute. 25. $\displaystyle θ=90°$ 27. $\displaystyle θ=60°$
29. $\displaystyle θ=114.6°$ 33. a. $\vecd{OP}=4\mathbf{\hat i} +5\mathbf{\hat j}, \; \vecd{OQ}=5\mathbf{\hat i} −7\mathbf{\hat j} $ b. $105.8°$ 35. $\displaystyle 68.33°$ 37. $\displaystyle a. \; 2\sqrt{2}; \; b. \; 109.47°$

D: Orthogonality

Exercise $\PageIndex{D}$

Determine whether the given vectors are orthogonal.

41) $\quad \vecs{a}=⟨x,y⟩, \quad \vecs{b}=⟨−y, x⟩$, where $x$ and $y$ are nonzero real numbers

42) $\quad \vecs{a}=⟨x, x⟩, \quad \vecs{b}=⟨−y, y⟩$, where $x$ and $y$ are nonzero real numbers

43) $\quad \vecs{a}=3\mathbf{\hat i} −\mathbf{\hat j} −2\mathbf{\hat k} , \quad \vecs{b}=−2\mathbf{\hat i} −3\mathbf{\hat j} +\mathbf{\hat k} $

44) $\quad \vecs{a}=\mathbf{\hat i} −\mathbf{\hat j} , \quad \vecs{b}=7\mathbf{\hat i} +2\mathbf{\hat j} −\mathbf{\hat k} $

45) Determine the real number $α$ such that vectors $\vecs a=2\mathbf{\hat i} +3\mathbf{\hat j} $ and $\vecs b=9\mathbf{\hat i} +α\mathbf{\hat j} $ are orthogonal.

46) Determine the real number $α$ such that vectors $\vecs a=−3\mathbf{\hat i} +2\mathbf{\hat j} $ and $\vecs b=2\mathbf{\hat i} +α\mathbf{\hat j} $ are orthogonal.

Determine which (if any) pairs of the following vectors are orthogonal.

47) $\quad\vecs u=⟨3,7,−2⟩, \quad \vecs v=⟨5,−3,−3⟩, \quad \vecs w=⟨0,1,−1⟩$

48) $\quad\vecs u=\mathbf{\hat i} −\mathbf{\hat k} , \quad \vecs v=5\mathbf{\hat j} −5\mathbf{\hat k} , \quad \vecs w=10\mathbf{\hat j} $

Answers to odd exercises:: 41. Orthogonal $ \qquad $ 43. Not orthogonal $ \qquad $ 45. $\displaystyle α=−6$ $ \qquad $
47. $\vecs u$ and $\vecs v$ are orthogonal; $\vecs v$ and $\vecs w$ are orthogonal, but $\vecs u$ and $\vecs w$ are not orthogonal.

E: Projections

Exercise $\PageIndex{E}$

For the following exercises, the vectors $\vecs u$ and $\vecs v$ are given.

a. Find the vector projection $\text{Proj}_\vecs{u}\vecs v$ of vector $\vecs v$ onto vector $\vecs u$ and the component of $\vecs v$ that is orthogonal to $\vecs u$, i.e., $\vecs v_\text{perp}$. Express your answers in component form.

b. Find the magnitude of the projection, $ \| \text{Proj}_\vecs{u}\vecs v \|, $ of vector $\vecs v$ onto vector $\vecs u$.

c. Find the vector projection $\text{Proj}_\vecs{v}\vecs u$ of vector $\vecs u$ onto vector $\vecs v$ and the component of $\vecs u$ that is orthogonal to $\vecs v$, i.e., $\vecs u_\text{perp}$. Express your answers in unit vector form.

d. Find the magnitude of the projection, $ \| \text{Proj}_\vecs{v}\vecs u \|, $ of vector $\vecs u$ onto vector $\vecs v$.

51. $\quad\vecs u=5\mathbf{\hat i} +2\mathbf{\hat j} , \quad \vecs v=2\mathbf{\hat i} +3\mathbf{\hat j} $

52. $\quad\vecs u=⟨4,4,0⟩, \quad \vecs v=⟨0,4,1⟩$

53. $\quad\vecs u=3\mathbf{\hat i} +2\mathbf{\hat k} , \quad \vecs v=2\mathbf{\hat j} +4\mathbf{\hat k} $

55. Consider the vectors $\vecs u=4\mathbf{\hat i} −3\mathbf{\hat j} $ and $\vecs v=3\mathbf{\hat i} +2\mathbf{\hat j} .$

a. Find the component form of vector $\text{Proj}_\vecs{u}\vecs v$ that represents the projection of $\vecs v$ onto $\vecs u$..

b. Write the decomposition $\vecs v=\vecs w+\vecs q$ of vector $\vecs v$ into the orthogonal components $\vecs w$ and $\vecs q$, where $\vecs w$ is the projection of $\vecs v$ onto $\vecs u$ and $\vecs q$ is the vector component of $\vecs v$ orthogonal to the direction of $\vecs u$. That is, $ \vecs q = \vecs v_\text{perp}$.

56. Consider vectors $\vecs u=2\mathbf{\hat i} +4\mathbf{\hat j} $ and $\vecs v=4\mathbf{\hat j} +2\mathbf{\hat k} .$

a. Find the component form of vector $\vecs w=\text{Proj}_\vecs{u}\vecs v$ that represents the projection of $\vecs v$ onto $\vecs u$.b. Write the decomposition $\vecs v=\vecs w+\vecs q$ of vector $\vecs v$ into the orthogonal components $\vecs w$ and $\vecs q$, where $\vecs w$ is the projection of $\vecs v$ onto $\vecs u$ and $\vecs q$ is a vector orthogonal to the direction of $\vecs u$.

Answers to odd exercises:: 51. a. $\text{Proj}_\vecs{u}\vecs v =⟨\frac{80}{29},\frac{32}{29}⟩$ and $\vecs v_\text{perp} = <-\frac{22}{29}, \frac{55}{29}>$;
b. $ \| \text{Proj}_\vecs{u}\vecs v \| =\frac{16}{\sqrt{29}} = \frac{16\sqrt{29}}{29};$
c. $\text{Proj}_\vecs{v}\vecs u = \frac{32}{13}\mathbf{\hat i} + \frac{48}{13}\mathbf{\hat j} $ and $\vecs u_\text{perp} = \frac{33}{13}\mathbf{\hat i} - \frac{22}{13}\mathbf{\hat j} $;
d. $ \| \text{Proj}_\vecs{v}\vecs u \| =\frac{16}{\sqrt{13}}=\frac{16\sqrt{13}}{13}$
53. a. $\text{Proj}_\vecs{u}\vecs v =⟨\frac{24}{13},0,\frac{16}{13}⟩$ and $\vecs v_\text{perp} = <-\frac{24}{13}, \frac{26}{13}, \frac{36}{13}>$;
b. $ \| \text{Proj}_\vecs{u}\vecs v \| =\frac{8}{\sqrt{13}}=\frac{8\sqrt{13}}{13}$
c. $\text{Proj}_\vecs{v}\vecs u =\frac{4}{5}\mathbf{\hat j} + \frac{8}{5}\mathbf{\hat k} $ and $\vecs u_\text{perp} = 3\mathbf{\hat i} - \frac{4}{5}\mathbf{\hat j} + \frac{2}{5}\mathbf{\hat k} $;
d. $ \| \text{Proj}_\vecs{v}\vecs u \| =\frac{\sqrt{80}}{5} = \frac{4\sqrt{5}}{5};$
55. a. $\text{Proj}_\vecs{u}\vecs v=⟨\frac{24}{25},−\frac{18}{25}⟩$;
b. $\vecs q = \vecs v_\text{perp} =⟨\frac{51}{25},\frac{68}{25}⟩$,
$\vecs v =\vecs w+\vecs q= \text{Proj}_\vecs{u}\vecs v + \vecs v_\text{perp} =⟨\frac{24}{25},−\frac{18}{25}⟩+⟨\frac{51}{25},\frac{68}{25}⟩$
So we have that, $\vecs v = ⟨\frac{24}{25},−\frac{18}{25}⟩+⟨\frac{51}{25},\frac{68}{25}⟩$.

F: Dot Product Applications

Exercise $\PageIndex{F}$

61. Find the work done by force $\vecs F=⟨5,6,−2⟩$ (measured in Newtons) that moves a particle from point $\displaystyle P(3,−1,0)$ to point $\displaystyle Q(2,3,1)$ along a straight line (the distance is measured in meters).

62. A sled is pulled by exerting a force of 100 N on a rope that makes an angle of $\displaystyle 25°$ with the horizontal. Find the work done in pulling the sled 40 m. (Round the answer to one decimal place.)

$This figure is an image of a person pulling a child on a sled. The rope for pulling the sled is represented by a vector and labeled “25 lb.” There is an angle between the rope vector and the horizontal ground of 20 degrees.$ 63. A father is pulling his son on a sled at an angle of $\displaystyle 20°$with the horizontal with a force of 25 lb (see the following image). He pulls the sled in a straight path of 50 ft. How much work was done by the man pulling the sled? (Round the answer to the nearest integer.)

64. A car is towed using a force of 1600 N. The rope used to pull the car makes an angle of 25° with the horizontal. Find the work done in towing the car 2 km. Express the answer in joules $\displaystyle (1J=1N⋅m)$ rounded to the nearest integer.

65. A boat sails north aided by a wind blowing in a direction of $\displaystyle N30°E$ with a magnitude of 500 lb. How much work is performed by the wind as the boat moves 100 ft? (Round the answer to two decimal places.)

66. Vector $\vecs p=⟨150,225,375⟩$ represents the price of certain models of bicycles sold by a bicycle shop. Vector $\vecs n=⟨10,7,9⟩$ represents the number of bicycles sold of each model, respectively. Compute the dot product $\vecs p ⋅ \vecs n$ and state its meaning.

67. Two forces $\vecs F_1$ and $\vecs F_2$ are represented by vectors with initial points that are at the origin. The first force has a magnitude of 20 lb and the terminal point of the vector is point $\displaystyle P(1,1,0)$. The second force has a magnitude of 40 lb and the terminal point of its vector is point $\displaystyle Q(0,1,1)$. Let $\vecs F$ be the resultant force of forces $\vecs F_1$ and $\vecs F_2$.

a. Find the magnitude of $\vecs F$. (Round the answer to one decimal place.)

Answers to odd exercises:: 61. $\displaystyle 17N⋅m$
63. 1175 ft⋅lb
65. $25000\sqrt{3}$ ft-lbs $\approx 43,301.27$ ft-lbs
Vector representing the wind: $\vecs w = 500\cos 60^{\circ} \mathbf{\hat i} + 500\sin 60^{\circ} \mathbf{\hat j}$
Vector representing the displacement to the north: $\vecs d = 100 \mathbf{\hat j}$
Work done by the wind: $W = \vecs w \cdot \vecs d = 25000\sqrt{3}$ ft-lbs $\approx 43,301.27$ ft-lbs
67. $\displaystyle a. ∥F_1+F_2∥=52.9$ lb;