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Mathematics LibreTexts

11.3E: Exercises for The Dot Product

 

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For exercises 1-4, the vectors \(\vec{u}\) and \(\vec{v}\) are given.  Calculate the dot product \(\vec{u}\cdot\vec{v}\).

1) \(\quad \vec{u}=⟨3,0⟩, \quad \vec{v}=⟨2,2⟩\)

Answer:
\(\vec{u}\cdot\vec{v}=6\)

2) \(\quad \vec{u}=⟨3,−4⟩, \quad \vec{v}=⟨4,3⟩\)

3) \(\quad \vec{u}=⟨2,2,−1⟩, \quad \vec{v}=⟨−1,2,2⟩\)

Answer:
\(\vec{u}\cdot\vec{v}=0\)

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

 

For exercises 5-8, the vectors \(\vec{a}, \,\vec{b}\), and \(\vec{c}\) are given.  Determine the vectors \((\vec{a}\cdot\vec{b})\vec{c}\) and \((\vec{a}⋅\vec{c})\vec{b}.\) Express the vectors in component form.

5) \(\quad \vec{a}=⟨2,0,−3⟩, \quad \vec{b}=⟨−4,−7,1⟩, \quad \vec{c}=⟨1,1,−1⟩\)

Solution: \(\displaystyle (a⋅b)c=⟨−11,−11,11⟩; (a⋅c)b=⟨−20,−35,5⟩\)

6) \(\quad \vec{a}=⟨0,1,2⟩, \quad \vec{b}=⟨−1,0,1⟩, \quad \vec{c}=⟨1,0,−1⟩\)

7) \(\quad \vec{a}=\hat{\imath}+\hat{\jmath}, \quad \vec{b}=\hat{\imath}−\hat{k}, \quad \vec{c}=\hat{\imath}−2\hat{k}\)

Solution: \(\displaystyle (a⋅b)c=⟨1,0,−2⟩; (a⋅c)b=⟨1,0,−1⟩\)

8) \(\quad \vec{a}=\hat{\imath}−\hat{\jmath}+\hat{k}, \quad \vec{b}=\hat{\jmath}+3\hat{k}, \quad \vec{c}=−\hat{\imath}+2\hat{\jmath}−4\hat{k}\)

 

For exercises 9-12, two vectors are given.

a. Find the measure of the angle \(θ\) between these two vectors.  Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly.

b. Is \(θ\) an acute angle?

9) [T] \(\quad \vec{a}=⟨3,−1⟩, \quad \vec{b}=⟨−4,0⟩\)

Solution: \(\displaystyle a. θ=2.82\)rad; \(\displaystyle b. θ\) is not acute.

10) [T] \(\quad \vec{a}=⟨2,1⟩, \quad \vec{b}=⟨−1,3⟩\)

11) \(\quad \vec{u}=3i, \quad \vec{v}=4\hat{\imath}+4\hat{\jmath}\)

Solution: \(\displaystyle a. θ=\frac{π}{4}\)rad; \(\displaystyle b. θ\) is acute.

12) \(\quad \vec{u}=5i, \quad \vec{v}=−6\hat{\imath}+6\hat{\jmath}\)

 

For exercises 13-18, find the measure of the angle between the three-dimensional vectors \(\vec{a}\) and \(\vec{b}\).  Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly.

13) \(\quad \vec{a}=⟨3,−1,2⟩, \quad \vec{b}=⟨1,−1,−2⟩\)

Solution: \(\displaystyle θ=\frac{π}{2}\)

14) \(\quad \vec{a}=⟨0,−1,−3⟩, \quad \vec{b}=⟨2,3,−1⟩\)

15) \(\quad \vec{a}=i+j, \quad \vec{b}=j−k\)

Solution: \(\displaystyle θ=\frac{π}{3}\)

16) \(\quad \vec{a}=i−2j+k, \quad \vec{b}=i+j−2k\)

17) [T] \(\quad \vec{a}=3i−j−2k, \quad \vec{b}=v+w,\) where \(\quad \vec{v}=−2i−3j+2k\) and \(\vec{w}=i+2k\)

Solution: \(\displaystyle θ=2\)rad

18) [T] \(\quad \vec{a}=3i−j+2k, \quad \vec{b}=v−w,\) where \(\quad \vec{v}=2i+j+4k\) and \(\vec{w}=6i+j+2k\)

 

For exercises 19-22, determine whether the given vectors are orthogonal.

19) \(\quad \vec{a}=⟨x,y⟩, \quad \vec{b}=⟨−y, x⟩\), where \(x\) and \(y\) are nonzero real numbers

Solution: Orthogonal

20) \(\quad \vec{a}=⟨x, x⟩, \quad \vec{b}=⟨−y, y⟩\), where \(x\) and \(y\) are nonzero real numbers

21) \(\quad \vec{a}=3i−j−2k, \quad \vec{b}=−2i−3j+k\)

Solution: Not orthogonal

22) \(\quad \vec{a}=i−j, \quad \vec{b}=7i+2j−k\)

 

23) Find all two-dimensional vectors a orthogonal to vector \(\displaystyle b=⟨3,4⟩.\) Express the answer in component form.

Solution: \(\displaystyle a=⟨−\frac{4α}{3},α⟩,\) where \(\displaystyle α≠0\) is a real number

24) Find all two-dimensional vectors \(\displaystyle a\) orthogonal to vector \(\displaystyle b=⟨5,−6⟩.\) Express the answer by using standard unit vectors.

25) Determine all three-dimensional vectors \(\displaystyle u\) orthogonal to vector \(\displaystyle v=⟨1,1,0⟩.\) Express the answer by using standard unit vectors.

Solution: \(\displaystyle u=−αi+αj+βk,\) where \(\displaystyle α\) and \(\displaystyle β\) are real numbers such that \(\displaystyle α^2+β^2≠0\)

26) Determine all three-dimensional vectors \(\displaystyle u\) orthogonal to vector \(\displaystyle v=i−j−k.\) Express the answer in component form.

27) Determine the real number \(\displaystyle α\) such that vectors \(\displaystyle a=2i+3j\) and \(\displaystyle b=9i+αj\) are orthogonal.

Solution: \(\displaystyle α=−6\)

28) Determine the real number \(\displaystyle α\) such that vectors \(\displaystyle a=−3i+2j\) and \(\displaystyle b=2i+αj\) are orthogonal.

29) [T] Consider the points \(\displaystyle P(4,5)\) and \(\displaystyle Q(5,−7)\).

a. Determine vectors \(\displaystyle \vec{OP}\) and \(\displaystyle \vec{OQ}\). 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.

Solution: \(\displaystyle a. \vec{OP}→=4i+5j, \vec{OQ}=5i−7j; b. 105.8°\)

30) [T] Consider points \(\displaystyle A(1,1), B(2,−7),\) and \(\displaystyle C(6,3)\).

a. Determine vectors \(\displaystyle \vec{BA}\) and \(\displaystyle \vec{BC}\). 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.

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

Solution: \(\displaystyle 68.33°\)

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

 

For the following exercises, determine which (if any) pairs of the following vectors are orthogonal.

33) \(\displaystyle u=⟨3,7,−2⟩, v=⟨5,−3,−3⟩, w=⟨0,1,−1⟩\)

Solution: \(\displaystyle u\) and \(\displaystyle v\) are orthogonal; \(\displaystyle v\) and \(\displaystyle w\) are orthogonal.

34) \(\displaystyle u=i−k, v=5j−5k, w=10j\)

35) Use vectors to show that a parallelogram with equal diagonals is a square.

36) Use vectors to show that the diagonals of a rhombus are perpendicular.

37) Show that \(\displaystyle u⋅(v+w)=u⋅v+u⋅w\) is true for any vectors \(\displaystyle u, v\), and \(\displaystyle w\).

38) Verify the identity \(\displaystyle u⋅(v+w)=u⋅v+u⋅w\) for vectors \(\displaystyle u=⟨1,0,4⟩, v=⟨−2,3,5⟩,\) and \(\displaystyle w=⟨4,−2,6⟩.\)

 

For the following problems, the vector \(\displaystyle u\) is given.

a. Find the direction cosines for the vector u.

b. Find the direction angles for the vector u expressed in degrees. (Round the answer to the nearest integer.)

39) \(\displaystyle u=⟨2,2,1⟩\)

Solution: \(\displaystyle a. cosα=\frac{2}{3},cosβ=\frac{2}{3},\) and \(\displaystyle cosγ=\frac{1}{3}; b. α=48°, β=48°,\) and \(\displaystyle γ=71°\)

40) \(\displaystyle u=i−2j+2k\)

41) \(\displaystyle u=⟨−1,5,2⟩\)

Solution: \(\displaystyle a. cosα=−\frac{1}{\sqrt{30}},cosβ=\frac{5}{\sqrt{30}},\) and \(\displaystyle cosγ=\frac{2}{\sqrt{30}}; b. α=101°, β=24°,\) and \(\displaystyle γ=69°\)

42) \(\displaystyle u=⟨2,3,4⟩\)

43) Consider \(\displaystyle u=⟨a,b,c⟩\) a nonzero three-dimensional vector. Let \(\displaystyle cosα, cosβ,\) and \(\displaystyle cosγ\) be the directions of the cosines of \(\displaystyle u\). Show that \(\displaystyle cos^2α+cos^2β+cos^2γ=1.\)

44) Determine the direction cosines of vector \(\displaystyle u=i+2j+2k\) and show they satisfy \(\displaystyle cos^2α+cos^2β+cos^2γ=1.\)

 

of vector v into the orthogonal components w and q, where w is the projection of v onto u and q is a vector orthogonal to the direction of u.

For the following exercises, the vectors \(\displaystyle u\) and \(\displaystyle v\) are given.

a. Find the vector projection \(\displaystyle w=proj_uv\) of vector \(\displaystyle v\) onto vector \(\displaystyle u\). Express your answer in component form.

b. Find the scalar projection \(\displaystyle comp_uv\) of vector \(\displaystyle v\) onto vector \(\displaystyle u\).

45) \(\displaystyle u=5i+2j, v=2i+3j\)

Solution: \(\displaystyle a. w=⟨\frac{80}{29},\frac{32}{29}⟩; b. comp_uv=\frac{16}{\sqrt{29}}\)

46) \(\displaystyle u=⟨−4,7⟩, v=⟨3,5⟩\)

47) \(\displaystyle u=3i+2k, v=2j+4k\)

Solution: \(\displaystyle a. w=⟨\frac{24}{13},0,\frac{16}{13}⟩; b. comp_uv=\frac{8}{\sqrt{13}}\)

48) \(\displaystyle u=⟨4,4,0⟩, v=⟨0,4,1⟩\)

49) Consider the vectors \(\displaystyle u=4i−3j\) and \(\displaystyle v=3i+2j.\)

a. Find the component form of vector \(\displaystyle w=proj_uv\) that represents the projection of \(\displaystyle v\) onto \(\displaystyle u\).

b. Write the decomposition \(\displaystyle v=w+q\) of vector \(\displaystyle v\) into the orthogonal components \(\displaystyle w\) and \(\displaystyle q\), where \(\displaystyle w\) is the projection of \(\displaystyle v\) onto \(\displaystyle u\) and \(\displaystyle q\) is a vector orthogonal to the direction of \(\displaystyle u\).

Solution: \(\displaystyle a. w=⟨\frac{24}{25},−\frac{18}{25}⟩; b. q=⟨\frac{51}{25},\frac{68}{25}⟩, v=w+q=⟨\frac{24}{25},−\frac{18}{25}⟩+⟨\frac{51}{25},\frac{68}{25}⟩\)

50) Consider vectors \(\displaystyle u=2i+4j\) and \(\displaystyle v=4j+2k.\)

a. Find the component form of vector \(\displaystyle w=proj_uv\) 0that represents the projection of \(\displaystyle v\) onto \(\displaystyle u\).

b. Write the decomposition \(\displaystyle v=w+q\) of vector \(\displaystyle v\) into the orthogonal components \(\displaystyle w\) and \(\displaystyle q\), where \(\displaystyle w\) is the projection of \(\displaystyle v\) onto \(\displaystyle u\) and \(\displaystyle q\) is a vector orthogonal to the direction of \(\displaystyle u\).

 

51) 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.

Solution: \(\displaystyle a. 2\sqrt{2}; b. 109.47°\)

52) [T] 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.

53) Find the work done by force \(\displaystyle F=⟨5,6,−2⟩\) (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).

Solution: \(\displaystyle 17N⋅m\)

54) [T] 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.)

55) [T] 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.)

Solution: 1175 ft⋅lb

56) [T] 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.

57) [T] 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.)

Solution: \(\displaystyle 4330.13 ft-lb\)

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

59) [T] Two forces \(\displaystyle F_1\) and \(\displaystyle F_2\) 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 F be the resultant force of forces \(\displaystyle F_1\) and \(\displaystyle F_2\).

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

b. Find the direction angles of \(\displaystyle F\). (Express the answer in degrees rounded to one decimal place.)

Solution: \(\displaystyle a. ∥F_1+F_2∥=52.9\) lb; b. The direction angles are \(\displaystyle α=74.5°,β=36.7°,\) and \(\displaystyle γ=57.7°.\)

60) [T] Consider \(\displaystyle r(t)=⟨cost,sint,2t⟩\) the position vector of a particle at time \(\displaystyle t∈[0,30]\), where the components of \(\displaystyle r\) are expressed in centimeters and time in seconds. Let \(\displaystyle \vec{OP}\) be the position vector of the particle after 1 sec.

a. Show that all vectors \(\displaystyle \vec{PQ}\), where \(\displaystyle Q(x,y,z)\) is an arbitrary point, orthogonal to the instantaneous velocity vector v(1) of the particle after 1 sec, can be expressed as \(\displaystyle \vec{PQ}=⟨x−cos1,y−sin1,z−2⟩\), where \(\displaystyle xsin1−ycos1−2z+4=0.\) The set of point Q describes a plane called the normal plane to the path of the particle at point P.

b. Use a CAS to visualize the instantaneous velocity vector and the normal plane at point P along with the path of the particle.

 

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY 3/0 license. Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191.

Exercises and LaTeX edited by Paul Seeburger