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# 5.3E: Excersies

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## Exercise $$\PageIndex{1}$$

For the following exercises, the vectors $$\displaystyle u$$ and $$\displaystyle v$$ are given. Calculate the dot product $$\displaystyle u⋅v$$.

1) $$\displaystyle u=⟨3,0⟩, v=⟨2,2⟩$$

6

2) $$\displaystyle u=⟨3,−4⟩, v=⟨4,3⟩$$

3) $$\displaystyle u=⟨2,2,−1⟩, v=⟨−1,2,2⟩$$

0

4) $$\displaystyle u=⟨4,5,−6⟩, v=⟨0,−2,−3⟩$$

## Exercise $$\PageIndex{2}$$

For the following exercises, the vectors $$\displaystyle u$$ and $$\displaystyle v$$ are given. Calculate the dot product $$\displaystyle u⋅v$$.

1) $$\displaystyle u=⟨3,0⟩, v=⟨2,2⟩$$

6

2) $$\displaystyle u=⟨3,−4⟩, v=⟨4,3⟩$$

3) $$\displaystyle u=⟨2,2,−1⟩, v=⟨−1,2,2⟩$$

0

4) $$\displaystyle u=⟨4,5,−6⟩, v=⟨0,−2,−3⟩$$

## Exercise $$\PageIndex{3}$$

For the following exercises, the vectors $$\displaystyle a, b$$, and $$\displaystyle c$$ are given. Determine the vectors $$\displaystyle (a⋅b)c$$ and $$\displaystyle (a⋅c)b.$$ Express the vectors in component form.

5) $$\displaystyle a=⟨2,0,−3⟩, b=⟨−4,−7,1⟩, c=⟨1,1,−1⟩$$

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

6) $$\displaystyle a=⟨0,1,2⟩, b=⟨−1,0,1⟩, c=⟨1,0,−1⟩$$

7) $$\displaystyle a=i+j, b=i−k, c=i−2k$$

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

8) $$\displaystyle a=i−j+k, b=j+3k, c=−i+2j−4k$$

## Exercise $$\PageIndex{4}$$

For the following exercises, the two-dimensional vectors $$\displaystyle a$$ and $$\displaystyle b$$ are given.

a. Find the measure of the angle $$\displaystyle θ$$ between a and b. Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly.

b. Is $$\displaystyle θ$$ an acute angle?

9) [T] $$\displaystyle a=⟨3,−1⟩, b=⟨−4,0⟩$$

$$\displaystyle a. θ=2.82$$rad; $$\displaystyle b. θ$$ is not acute.

10) [T] $$\displaystyle a=⟨2,1⟩, b=⟨−1,3⟩$$

11) $$\displaystyle u=3i, v=4i+4j$$

$$\displaystyle a. θ=\frac{π}{4}$$rad; $$\displaystyle b. θ$$ is acute.

12) $$\displaystyle u=5i, v=−6i+6j$$

## Exercise $$\PageIndex{5}$$

For the following exercises, find the measure of the angle between the three-dimensional vectors $$\displaystyle a$$ and $$\displaystyle b$$. Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly.

13) $$\displaystyle a=⟨3,−1,2⟩, b=⟨1,−1,−2⟩$$

$$\displaystyle θ=\frac{π}{2}$$

14) $$\displaystyle a=⟨0,−1,−3⟩, b=⟨2,3,−1⟩$$

15) $$\displaystyle a=i+j, b=j−k$$

$$\displaystyle θ=\frac{π}{3}$$

16) $$\displaystyle a=i−2j+k, b=i+j−2k$$

17) [T] $$\displaystyle a=3i−j−2k, b=v+w,$$ where $$\displaystyle v=−2i−3j+2k$$ and $$\displaystyle w=i+2k$$

$$\displaystyle θ=2$$rad

18) [T] $$\displaystyle a=3i−j+2k, b=v−w,$$ where $$\displaystyle v=2i+j+4k$$ and $$\displaystyle w=6i+j+2k$$

## Exercise $$\PageIndex{6}$$

For the following exercises determine whether the given vectors are orthogonal.

19) $$\displaystyle a=⟨x,y⟩, b=⟨−y,x⟩,$$ where x and y are nonzero real numbers

Orthogonal

20) $$\displaystyle a=⟨x,x⟩, b=⟨−y,y⟩,$$ where x and y are nonzero real numbers

21) $$\displaystyle a=3i−j−2k, b=−2i−3j+k$$

Not orthogonal

22) $$\displaystyle a=i−j, b=7i+2j−k$$

## Exercise $$\PageIndex{7}$$

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

$$\displaystyle a=⟨−\frac{4α}{3},α⟩,$$ 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.

$$\displaystyle u=−αi+αj+βk,$$ 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.

$$\displaystyle α=−6$$

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

## Exercise $$\PageIndex{8}$$

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.

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

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.

$$\displaystyle 68.33°$$

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.

## Exercise $$\PageIndex{9}$$

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

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

$$\displaystyle u$$ 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 $$\displaystyle w=⟨4,−2,6⟩.$$

## Exercise $$\PageIndex{10}$$

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) $$\displaystyle u=⟨2,2,1⟩$$

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

40) $$\displaystyle u=i−2j+2k$$

41) $$\displaystyle u=⟨−1,5,2⟩$$

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

42) $$\displaystyle u=⟨2,3,4⟩$$

## Exercise $$\PageIndex{11}$$

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 $$\displaystyle cos^2α+cos^2β+cos^2γ=1.$$

## Exercise $$\PageIndex{12}$$

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

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

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

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

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

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 $$\displaystyle u$$.

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

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 $$\displaystyle u$$.

## Exercise $$\PageIndex{13}$$

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. $$\displaystyle a. 2\sqrt{2}; b. 109.47°$$

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).

$$\displaystyle 17N⋅m$$

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.) 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.)

$$25000\sqrt{3}$$ ft-lbs $$\approx 43,301.27$$ ft-lbs
Solution:
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

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.

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