· Web viewAn Airbus A320 jet maintains a constant airspeed of 500 mph headed due north. The jet stream is 100 mph in the northeasterly direction (N45 E). Express the velocity v

Pre-AP PrecalculusModule 10

Vectors and Parametric Equations

10.1 VectorsA vector is a quantity that has both magnitude (measurement) and direction. It’s represented by a directed line segment with an arrowhead on one end, where the length of the segment represents magnitude and the arrowhead represents direction. If an arrowhead is applied to point Q on PQ, then the vector PQ has initial point P and terminal point Q. If vector v has the same magnitude as vector PQ , then we say v=PQ.

Two vectors v and w are equal if they have the same magnitude and direction regardless of the position of their initial points.

v=−w if vectors v and w have the same magnitude but opposite directions.

Note that in a typed format, when using a single letter to denote the name of a vector, we use a boldface letter, but in handwritten work, we do draw an arrow above the letter.

When adding vectors together, we position each vector so that its initial point coincides with the terminal point of the preceding vector. The first vector can be placed anywhere. The sum (or resultant) is the vector that starts at the first initial point and ends at the final terminal point.

Properties

Commutative Property: v+w=w+v Associative Property: u+ ( v+w )=(u+v )+w v+0=0+v=v v+ (−v )=0 If α is a scalar and v is a vector, the scalar multiple α v is defined as:

o If α>0, α v is the vector whose magnitude is α times the magnitude of v and whose direction is the same as v.

o If α<0, α v is the vector whose magnitude is |α| times the magnitude of v and whose direction is opposite that of v.

o If α=0 or if v=0, then α v=0. (α+β ) v=α v+β v α (v+w )=α v+α w α (β v )=(αβ)v

Example 1 – Use the given vectors to graph each of the following vectors: (a) v−w , (b) 2v+3w , and (c) 2v−w+u.

Sullivan & Sullivan – Section 9.4

10.1 VectorsWe use the symbol ‖v‖ to represent the magnitude of a vector v. Since ‖v‖ equals the length of a directed line segment,

‖v‖≥0 ‖v‖=0 if and only if v=0 ‖v‖=‖−v‖ ‖α v‖=|α|‖v‖

A vector u for which ‖u‖=1 is called a unit vector, and is denoted as u.

i= ⟨1,0 ⟩ is the unit vector in the direction of the positive x-axis, and j=⟨0,1 ⟩ is the unit vector in the direction of the positive y-axis. If a>0 and b>0 are both scalar values, then

a i is a vector with magnitude a in the direction of i b j is the vector with magnitude b in the direction of j −a i is the vector with magnitude a in the opposite direction of i −b j is the vector with magnitude b in the direction of j

v=a i+b j=⟨a ,b ⟩ is the component form of the resultant vector that extends a units horizontally (to the left if a<0 and to the right if a>0) and b units vertically (up if b>0 or down if b<0). a and b are the components of v.

A vector that has an initial point at the origin is called a position vector, and as a consequence the terminal point of v=⟨a , b ⟩ is (a ,b ). A vector whose initial point is not at the origin is called a displacement vector.

Suppose that v is a displacement vector with initial point P1=(x1 , y1 ), not necessarily the origin, and

terminal point P2=(x2 , y2). If v=P1P2, then v is equal to the position vector ⟨ x2−x1, y2− y1 ⟩.

Example 2 – Find the displacement vector P1P2 if P1=(−1,2 ) and P2=( 4,6 ).

If v=⟨a1 , b1 ⟩ and w= ⟨a2 , b2 ⟩ , then:

v=w if and only if a1=a2 and b1=b2. v±w=⟨a1±a2, b1±b2 ⟩=(a1±a2 ) i+(b1±b2 ) j α v= ⟨α a1 , α b1 ⟩=α a1 i+α b1 j


10.1 Vectors


10.1 Vectors

Just like with polar complex notation, if v=⟨a ,b ⟩, then ‖v‖=√a2+b2 and the direction of v is θv=tan−1 b

a or

θv=tan−1 b

a+180 °, depending on which quadrant the corresponding position vector would terminate in.

Likewise, a vector w with magnitude a in the direction θw can be written as w= ⟨acosθw ,b sin θw ⟩.

Example 3 – If v=⟨2,3 ⟩ and w= ⟨3 ,−4 ⟩, find (a) v+w and (b) v−w .

Example 4 – If v=⟨2,3 ⟩ and w= ⟨3 ,−4 ⟩, find (a) 3v , (b) 2v−3w, and (c) ‖v‖.

For any nonzero vector v, the vector v=v

‖v‖ is a unit vector that has the same direction as v. As a

consequence, v=‖v‖v .

Example 5 – Find a unit vector in the same direction as v=⟨ 4 ,−3 ⟩.


10.1 VectorsExample 6 – A ball is thrown with an initial speed of 25 mph in a direction that makes an angle of 30° with the positive x-axis. Express the velocity vector v in terms of i and j . What is the initial speed in the horizontal direction? What is the initial speed in the vertical direction?

Example 7 – Find the direction angles of (a) v=⟨4 ,−4 ⟩ and (b) w= ⟨−3 ,−3 √3 ⟩.

Example 8 – A Boeing 737 aircraft maintains a constant airspeed of 500 mph headed due south. The jet stream is 80 mph in the northeasterly direction (N45°E). (a) Express the velocity va of the 737 relative to the air and the velocity vw of the jet stream in terms of i and j . (b) Find the velocity of the 737 relative to the ground. (c) Find the actual speed and direction of the 737 relative to the ground.


10.1 VectorsExample 9 – Two movers require a magnitude of force of 300 pounds to push a piano up a ramp inclined at an angle 20° from the horizontal. How much does the piano weigh?

An object is said to be in static equilibrium if the object is at rest and the sum of all forces acting on the object is zero.

Example 10 – A box of supplies that weighs 1200 pounds is suspended by two cables attached to the ceiling, one forming an angle of 30° with the ceiling and the other forming an angle of 45°. What are the tensions in the two cables?

Complete the following exercises on a separate sheet of paper.

In exercises 1 – 8, use the figure to determine whether the given statement is true or false.

1. A+B=F

2. K+G=F

3. C=D−E+F

4. G+H+E=D

5. E+D=G+H

6. H−C=G−F

7. A+B+K+G=0

8. A+B+C+H+G=0


A

B

C

DE

F

G

K

H

10.1 VectorsIn exercises 9 – 12, find the displacement vector PQ in the form ⟨a ,b ⟩.

9. P= (0,0 );Q=(−3 ,−5 )

10. P= (−3,2 );Q=(6,5 )

11. P= (−1,4 ) ;Q=(6,2 )

12. P= (1,1 );Q= (2,2 )

In exercises 13 – 18, find ‖v‖.

13. v=3 i−4 j

14. v=−5 i+12 j

15. v=i− j

16. v=−i− j

17. v=−2 i+3 j

18. v=6 i+2 j

In exercises 19 – 24, find each quantity if v=3 i−5 j and w=−2 i+3 j.

19. 2v+3w

20. 3v−2w

21. ‖v−w‖

22. ‖v+w‖

23. ‖v‖−‖w‖

24. ‖v‖+‖w‖

In exercises 25 – 28, find the unit vector in the same direction as v.

25. v=3 i−4 j

26. v=−5 i+12 j

27. v=i− j

28. v=2 i− j

29. Find a vector v whose magnitude is 4 and whose component in the i direction is twice the component in the

j direction.

30. Find a vector v whose magnitude is 3 and whose component in the i direction is equal to the component in the j direction.

31. If v=2 i− j and w=x i+3 j, find all numbers x for which ‖v+w‖=5.

32. If P= (−3,1 ) and Q= (x ,4 ), find all numbers x such that the vector represented by PQ has length 5.

In exercises 33 – 36, write the vector v in the form ⟨a ,b ⟩, given its magnitude ‖v‖ and the angle α it makes with the positive x-axis.

33. ‖v‖=8 , α=45 °

34. ‖v‖=14 ;α=120 °

35. ‖v‖=3; α=240°

36. ‖v‖=15; α=315°

In exercises 37 – 40, find the magnitude and direction angle of v for each vector.

37. v=i+√3 j

38. v=−5 i−5 j

39. v=6 i−4 j

40. v=−i+3 j

41. Tw forces F1 and F2 of magnitudes 30 newtons (N) and 70 N act on an object at angles of 45° and 120°,

respectfully. Find the direction and magnitude of the resultant force; i.e. F1+F2.


10.1 Vectors42. An Airbus A320 jet maintains a constant airspeed of 500 mph headed due north. The jet stream is 100 mph

in the northeasterly direction (N45°E).

a. Express the velocity va of the A320 relative to the air and the velocity vw of the jet stream in terms of i and j .

b. Find the velocity of the A320 relative to the ground.

c. Find the actual speed and direction of the A320 relative to the ground.

43. An airplane has an airspeed of 600 km/hr bearing S30°E. The wind velocity is 40 km/hr in the direction S45°E. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction?

44. A magnitude of 1200 pounds of force is required to prevent a car from rolling down a hill whose incline is 15° to the horizontal. What is the weight of the car?

45. The pilot of an aircraft wishes to head direction east but is faced with a wind speed of 40 mph from the northwest. If the pilot maintains an airspeed of 250 mph, what compass heading should be maintained to head directly east? What is the actual speed of the aircraft?

46. A weight of 800 pounds is suspended from two cables, which makes angles of 35° and 50° with the ceiling. What are the tensions in the two cables?

47. At a county fair truck pull, two pickup trucks are attached to the back end of a monster truck. One of the pickups pulls with a force of 2000 pounds and the other pulls with a force of 3000 pounds with an angle of 45° between them. With how much force must the monster truck pull in order to remain unmoved? (Hint: Find the resultant force of the two trucks.)

48. A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighbor’s tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a 40° angle between the forces.

a. Assuming the farmer’s estimate of a needed 6-ton force is correct, will the farmer be successful in removing the stump? Explain.

b. Had the farmer arranged the tractors with a 25° angle between the forces, would he have been successful in removing the stump? Explain.


10.2 The Dot ProductIf v=a1 i+b1 j and w=a2 i+b2 j, are two vectors, then the dot or scalar product v ∙ w=a1a2+b1b2.

Example 1 – If v=⟨2 ,−3 ⟩ and w= ⟨5,3 ⟩, find: (a) v ∙w (b) w ∙ v, (c) v ∙ v , (d) w ∙w , (e) ‖v‖, and (f) ‖w‖.

If u, v, and w are vectors, then

u ∙ v=v ∙u u ∙ (v+w )=u ∙ v+u ∙w v ∙ v=‖v‖2

0 ∙ v=0

If u and v are two nonzero vectors, the angle θ, 0≤θ≤π , between u and v is determined by the formula u ∙ v=‖u‖‖v‖cosθ. Unlike matrix addition that required the vectors be positioned head-to-tail, matrix multiplication requires the vectors be placed tail-to-tail.

Example 2 – Find the angle θ between u=4 i−3 j and v=2 i+5 j.

Example 3 – Find the angle θ between v=⟨3 ,−1 ⟩ and w= ⟨6 ,−2 ⟩.

Sullivan & Sullivan – Section 9.5Foerster – Section 10.4

10.2 The Dot ProductTwo vectors v and w are orthogonal if and only if v ∙w=0.

Example 4 – Show that v=⟨2 ,−1 ⟩ and w= ⟨3,6 ⟩ are orthogonal.

In the last section, we decomposed vectors into their horizontal and vertical components. In fact, there are times when we will wish to decompose vectors into orthogonal components not aligned to the horizontal and vertical. For instance, looking back at Example 9 in 10.1, movers were moving a piano up the ramp. We decomposed the weight of the piano (a force vector point straight down) into the force vector perpendicular to the ramp’s surface (friction) and the force vector running parallel to the ramp’s surface.

The component of a vector v in the direction of a second vector w is known as the vector projection. The magnitude of the vector projection is known as the scalar projection. If v and w have an angle θ between them when placed tail-to-tail, then the scalar projection of v onto w (or in the direction of w) is ‖p‖=‖v‖cosθ. For the vector projection, we must apply a direction to the scalar projection without changing the magnitude. Since the direction is determined by w, we can multiply the scalar projection by

the unit vector in the direction of w, namely w

‖w‖. Thus the vector projection p=w‖v‖cos θ

‖w‖ . The

numerator almost looks like v ∙w=‖v‖‖w‖cosθ, and by multiplying by ‖w‖‖w‖, we get the equivalent formula

p= v ∙w‖w‖2

w. The component orthogonal to the vector projection is equal to v−p.

Example 5 – Find the vector projection of v=⟨1,3 ⟩ onto w= ⟨1,1 ⟩. Decompose v into two vectors, v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.


10.2 The Dot ProductExample 6 – A wagon with two small children as occupants that weighs 100 pounds is on a hill with a grade of 20°. What is the magnitude of the force that is required to keep the wagon from rooling down the hill?

In elementary physics, the work W done by a constant force F in moving an object from a point A to a point B is defined as W=(magnitude of force ) (distance )=‖F‖‖AB‖. This definition presupposes that the force is being done in the same direction as the motion, but if the force is applied at an angle θ to the direction of the motion, then W=F ∙ AB.

Example 7 – A girl pulls a wagon along level ground with a force of 50 pounds. How much work is done in moving the wagon 100 feet if the handle makes an angle of 30° with the horizontal?


In exercises 1 – 16, (a) find the dot product v ∙ w; (b) find the angle between v and w; and (c) state whether the vectors are parallel, orthogonal, or neither.

1. v=⟨1 ,−1 ⟩, w= ⟨1,1 ⟩

2. v=⟨1,1 ⟩, w= ⟨−1,1 ⟩

3. v=⟨2,1 ⟩, w= ⟨1,−2 ⟩

4. v=⟨2,2 ⟩, w= ⟨1,2 ⟩

5. v=⟨√3 ,−1 ⟩, w= ⟨1,1 ⟩

6. v=⟨1,√3 ⟩, w= ⟨1,−1 ⟩

7. v=⟨3,4 ⟩ , w= ⟨−6 ,−8 ⟩

8. v=⟨3 ,−4 ⟩, w= ⟨9 ,−12 ⟩

9. v=4 i, w= j

10. v=i, w=−3 j

11. Find a so that the vectors ⟨1 ,−a ⟩ and ⟨2,3 ⟩ are orthogonal.


10.2 The Dot Product12. Find b so that the vectors ⟨1,1 ⟩ and ⟨1 , b ⟩ are orthogonal.

In exercises 13 – 18, decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w .

13. v=⟨2 ,−3 ⟩, w= ⟨1,−1 ⟩

14. v=⟨−3,2 ⟩, w= ⟨2,1 ⟩

15. v=⟨1 ,−1 ⟩, w= ⟨−1 ,−2 ⟩

16. v=⟨2 ,−1 ⟩, w= ⟨1,−2 ⟩

17. v=⟨3,1 ⟩, w= ⟨−2 ,−1 ⟩

18. v=⟨1 ,−3 ⟩, w= ⟨4 ,−1 ⟩

19. Find the work done by a force of 3 pounds acting in the direction 60° to the horizontal in moving an object 6 feet from (0,0 ) to (6,0 ).

20. A wagon is pulled horizontally by exerting a force of 20 pounds on the handle at an angle of 30° with the horizontal. How much work is done in moving the wagon 100 feet?

21. The amount of energy collected by a solar panel depends on the intensity of the sun’s rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of the sun’s rays. Let the vector A represent the area, in square centimeters, whose direction is the orientation of a solar panel. The total number of watts collected by the panel is given by W=|I ∙ A|. Suppose that I=⟨−0.02 ,−0.01 ⟩ and A=⟨300,400 ⟩.

a. Find ‖I‖ and ‖A‖ and interpret the meaning of each.

b. Compute W and interpret its meaning.

c. If the solar panel is to collect the maximum number of watts, what must be true about I and A?

22. Let the vector R represent the amount of rainfall, in inches, whose direction is the inclination of the rain to a rain gauge. Let the vector A represent the area, in square inches, whose direction is the orientation of the opening of the rain gauge. The volume of rain collected in the gauge, in cubic inches, is given by V=|R∙ A|, even when the rain falls in a slanted direction or the gauge is not perfectly vertical. Suppose that R=⟨0.75 ,−1.75 ⟩ and A=⟨0.3,1 ⟩.

a. Find ‖R‖ and ‖A‖ and interpret the meaning of each.

b. Compute V and interpret its meaning.

c. If the gauge is to collect the maximum volume of rain, what must be true about R and A?

23. A Toyota Sienna with a gross weight of 5300 pounds is parked on a street with an 8° grade. Find the magnitude of the force required to keep the Sienna from rolling down the hill. What is the magnitude of the force perpendicular to he hill?

24. A Pontiac Bonneville with a gross weight of 4500 pounds is parked on a street with a 10° grade. Find the magnitude of the force required to keep the Bonneville from rolling down the hill. What is the magnitude of the force perpendicular to the hill?

25. Billy any Timmy are using a ramp to load furniture into a truck. While rolling a 250-pound piano up the ramp, they discover that the truck is too full of other furniture for the piano to fit. Timmy holds the piano in


10.2 The Dot Productplace on the ramp while Billy repositions other items to make room for it in the truck. If the angle of inclination of the ramp is 20°, how many pounds of force must Timmy exert to hold the piano in position?

26. A bulldozer exerts 1000 pounds of force to prevent a 5000-pound boulder from rolling down a hill. Determine the angle of inclination of the hill.


10.3 Vectors in SpaceCoordinates in three dimensional Cartesian space are of the form ( x , y , z ), where x=0 is the equation for the yz-plane, y=0 is the xz-plane, and z=0 is the xy-plane.

If P1=(x1 , y1 , z1) and P2=(x2 , y2 , z2 ) are two points in space, the distance d from P1 to P2 is

d=√(x2−x1 )2+( y2− y1 )2+( z2−z1 )2.

Example 1 – Find the distance from P1=(−1,3,2 ) to P2=(4 ,−2,5 ).

To represent vectors in space, we use unit vectors i , j and k , whose directions are along the positive x-axis, positive y-axis, and positive z-axis, respectively. If a position vector has a terminal point at (a ,b , c ), we can defined it as a i+b j+c k=⟨ a ,b ,c ⟩, where a, b, and c are the vector’s components. The displacement vector with initial point P1=(x1 , y1 , z1) and terminal point P2=(x2 , y2 , z2 ) is

P1P2=(x2−x1 ) i+ ( y2− y1 ) j+( z2−z1 ) k.

Example 2 – Find the displacement vector P1P2 if P1=(−1,2,3 ) and P2=( 4,6,2 ).

Let v=⟨a1 , b1 ,c1 ⟩ and w= ⟨a2 , b2 , c2 ⟩ be two vectors and let α be a scalar. Then:

v=w if and only if a1=a2 , b1=b2 ,and c1=c2. v ± w=(a1±a2 ) i+ (b1±b2 ) j+(c1+c2 ) k α v=(α a1 ) i+(αb❑1 ) j+ (α c1 ) k ‖v‖=√a12+b12+c12


10.3 Vectors in SpaceExample 3 – If v=⟨2,3 ,−2 ⟩ and w= ⟨3 ,−4,5 ⟩, find (a) v+ w and (b) v−w .

Example 4 – If v=⟨2,3 ,−2 ⟩ and w= ⟨3 ,−4,5 ⟩, find (a) 3 v , (b) 2 v−3 w, and (c) ‖v‖.

Recall v=v

‖v‖ is a unit vector in the direction of v, and as a consequence v=‖v‖v .

Example 5 – Find the unit vector in the same direction as ⟨2,−3 ,−6 ⟩.

If v=⟨a1 , b1 ,c1 ⟩ and w= ⟨a2 , b2 , c2 ⟩ with an angle θ between them, then v ∙ w=a1a2+b1b2+c1 c2=‖v‖‖w‖cosθ.


10.3 Vectors in SpaceExample 6 – If v=⟨2 ,−3,6 ⟩ and w= ⟨5,3 ,−1 ⟩, find (a) v ∙ w , (b) w ∙ v, (c) v ∙ v , (d) w ∙ w , (e) ‖v‖, and (f) ‖w‖.

If u, v, and w are vectors, then:

u ∙ v= v ∙ u u ∙ ( v+w )=u ∙ v+u ∙ w v ∙ v=‖v‖2

0 ∙ v=0

Example 7 – Find the angle θ between ⟨2,−3,6 ⟩ and ⟨2,5 ,−1 ⟩ when placed tail-to-tail.

A nonzero vector v in space can be described by specifying its magnitude and its three direction angles, α ,β , and γ. These direction angles are defined as:

α=¿ the angle between v and i, 0≤α≤π β=¿ the angle between v and j , 0≤ β≤π γ=¿ the angle between v and k , 0≤ γ ≤ π

If v=⟨a ,b , c ⟩ is a nonzero vector in space, the direction angles α ,β ,and γ obey cos α=a

‖v‖, cos β=b

‖v‖, and

cos γ= c‖v‖.

Example 8 – Find the direction angles of ⟨−3,2,−6 ⟩.


10.3 Vectors in SpaceIf α ,β , and γ are the direction angles of a nonzero vector in space, then cos2α+cos2β+cos2γ=1.

Example 9 – The vector v makes an angle of α= π3 with the positive x-axis, an angle of β=

π3 with the

positive y-axis, and an acute angle γ with the positive z-axis. Find γ.

It can be shown that v=‖v‖[ (cosα ) i+(cos β ) j+(cos γ ) k ].

Example 10 – Find the point that is 23 of the way from (5 ,−2,−6 ) to (11 ,−11 ,6 ).

Example 11 – Find the vector projection of ⟨ 3 ,5 ,−4 ⟩ onto ⟨1 ,−7 ,2 ⟩.


10.3 Vectors in SpaceComplete the following exercises on a separate sheet of paper.

In exercises 1 – 4, find the displacement vector PQ .

1. P= (0,0,0 ) and Q= (−3 ,−5,4 )

2. P= (3,2 ,−1 ) and Q= (5,6,0 )

3. P= (−3,2,0 ) and Q= (6,5 ,−1 )

4. P= (−1,4 ,−2 ) and Q= (6,2,2 )

In exercises 5 – 8, find ‖v‖.

5. v=⟨3 ,−6 ,−2 ⟩

6. v=⟨−6,12,4 ⟩

7. v=⟨−1 ,−1,1 ⟩

8. v=⟨6,2 ,−2 ⟩

In exercises 9 – 14, find each requested quantity if v=3 i−5 j+2 k and w=−2 i+3 j−2 k .

9. 2 v+3 w

10. 3 v−2 w

11. ‖v−w‖

12. ‖v+ w‖

13. ‖v‖−‖w‖

14. ‖v‖+‖w‖

In exercises 15 – 20, find the unit vector v in the same direction as v.

15. v=5 i

16. v=−2 j

17. v=3 i−6 j−2 k

18. v=−6 i+12 j+4 k

19. v=i+ j+k

20. v=2 i− j+k

In exercises 21 – 24, find the dot product v ∙ w and the angle between v and w .

21. v=i+ j, w=−i+ j−k

22. v=2 i+2 j−k , w=i+2 j+3 k

23. v=i+3 j+2 k , w=i− j+ k

24. v=3 i−4 j+k , w=6 i−8 j+2 k

In exercises 25 – 28, find the direction angles of each vector. Write each vector in the form v=‖v‖[ (cosα ) i+(cos β ) j+(cos γ ) k ].

25. v=⟨−6,12,4 ⟩

26. v=⟨1 ,−1,−1 ⟩

27. v=⟨0,1,1 ⟩

28. v=⟨2,3 ,−4 ⟩

In exercises 29 – 32, find the radius and center of each sphere.

29. x2+ y2+z2+2 x−2 z=2

30. x2+ y2+z2−4 x+4 y+2 z=0

31. x2+ y2+z2−4 x=0

32. 3 x2+3 y2+3 z2+6 x−6 y=3

33. Find the work done by a force of 3 newtons acting in the direction 2 i+ j+2 k in moving an object 2 meters from (0,0,0 ) to (0,2,0 ).

34. Find the work done in moving an object along a vector u=3 i+2 j−5 k if the applied force is F=2 i− j− k . Use meters for distance and newtons for force.


10.3 Vectors in Space35. Find the point that is 75% of the way from (−3,5,2 ) to (1 ,−3,14 ).

36. Find the point that is 13 of the way from (5,3 ,−7 ) to (11,0 ,−16 ).

37. Find the vector projection of ⟨5 ,−2,1 ⟩ onto ⟨8,2,−5 ⟩ .

38. Find the vector projection of ⟨7,2 ,−6 ⟩ onto ⟨−3,5 ,−1 ⟩ .


10.4 The Cross Product and Planar EquationsIf v=⟨a1 , b1 ,c1 ⟩ and w= ⟨a2 , b2 , c2 ⟩ are two vectors in space, then the cross product (or vector product)

v× w=(b1 c2−b2 c1 ) i−(a1c2−a2c1 ) j+(a1b2−a2b1 ) k .

Example 1 – If v=⟨2,3,5 ⟩ and w= ⟨1,2,3 ⟩, find v× w.

Example 2 – Calculate |2 31 2| and |A B C

2 3 51 2 3|.

The cross product of the vectors v=⟨a1 , b1 ,c1 ⟩ and w= ⟨a2 , b2 , c2 ⟩ is v× w=| i j ka1 b1 c1a2 b2 c2|.

Example 3 – If v=⟨2,3,5 ⟩ and w= ⟨1,2,3 ⟩, find (a) v× w, (b) w× v , (c) v× v, and (d) w × w.

Sullivan & Sullivan – Section 9.7Foerster – Sections 10.5 & 10.6

10.4 The Cross Product and Planar EquationsIf u , v , and w are vectors in space and if α is a scalar, then

u×u=0 u× v=−( v× u ) α (u× v )= (α u )× v=u× (α v ) u ( v+ w )= (u× v )+( u× v ) u× v is orthogonal to both u and v (and normal to the plane that contains u and v) ‖u× v‖=‖u‖‖v‖sinθ where θ is the angle between u and v when placed tail-to-tail.

‖u× v‖ is the area of the parallelogram having u ≠ 0 and v ≠ 0 as adjacent sides.

12‖u× v‖ is the area of the triangle having u ≠ 0 and v ≠ 0 as adjacent sides.

u× v=0 if and only if u and v are parallel.

Example 4 – Find a vector that is orthogonal to u=⟨3 ,−2,1 ⟩ and v=⟨−1,3 ,−1 ⟩.

Example 5 – Find the area of the parallelogram whose vertices are P1=(0,0,0 ), P2=(3 ,−2,1 ), P3=(−1,3 ,−1 ), and P4=(2,1,0 ).

The general form of the equation of a plane in space is Ax+By+Cz=D, where n=⟨ A , B ,C ⟩ is a normal vector to the plane.

Example 6 – Find the equation of the plane containing the point (3,5,7 ) with normal vector n=⟨11,2,13 ⟩.


10.4 The Cross Product and Planar EquationsExample 7 – Find two vectors n1 and n2 normal to the plane 7 x−3 y+8 z=−51.

Example 8 – Find an equation of the plane perpendicular to the segment connecting points P1=(3,8 ,−2 ) and P2=(7 ,−1,6 ) and passing through the point 30% of the way from point P1 to point P2.

Example 9 – Find a particular equation of the plane containing the points P1=(−5,5,5 ), P2=(−3,2,7 ), and P3=(1,12,6 ).


In exercises 1 – 4, find v× w and w× v .

1. v=⟨−1,3,2 ⟩, w= ⟨3 ,−2 ,−1 ⟩

2. v=⟨1 ,−4,2 ⟩ , w= ⟨3,2,1 ⟩

3. v=⟨3,1,3 ⟩, w= ⟨1,0 ,−1 ⟩

4. v=⟨2 ,−3,0 ⟩ , w= ⟨0,3 ,−2 ⟩


10.4 The Cross Product and Planar EquationsIn exercises 5 – 14, find each expression given that u=⟨2 ,−3,1 ⟩ , v= ⟨−3,3,2 ⟩, and w= ⟨1,1,3 ⟩.

5. v× w

6. w × u

7. v×4 w

8. −3 u×w

9. v ∙( v× w)

10. ( u× v ) ∙ w

11. v ∙(u× w)

12. ( u× v )×w

13. Find a vector orthogonal to both v and i+ j.

14. Find a vector orthogonal to both u and j+ k.

In exercises 15 – 16, find the area of the parallelogram with vertices P1 ,P2 ,P3 , and P4.

15. P1=(2,1,1 ) , P2=(2,3,1 ) ,P3=(−2,4,1 ) ,P4=(−2,6,1 )

16. P1=(−1,1,1 ) , P2=(−1,2,2 ), P3=(−3,4 ,−5 ) , P4=(−3,5 ,−4 )

In exercises 17 – 18, find the area of the triangle with vertices P1 ,P2 ,and P3.

17. P1=(0,0,0 ), P2=(2,3,1 ) , P3=(−2,4,1 ) 18. P1=(−2,0,2 ) , P2=(2,1 ,−1 ) , P3=(2 ,−1,2 )

In exercises 19 – 20, find two normal vectors to the plane, pointing in opposite directions.

19. 3 x+5 y−7 z=−13 20. 4 x−7 y+2 z=9

In exercises 21 -26, find a particular equation of the plane described.

21. Perpendicular to n=⟨3 ,−5,4 ⟩, containing the point (6 ,−7 ,−2 ).

22. Perpendicular to n=⟨−1,3 ,−2 ⟩, containing the point (4,7,5 ).

23. Perpendicular to the line segment connecting the points (3,8,5 ) and (11,2 ,−3 ) and passing through the midpoint of the segment.

24. Parallel to the plane 3 x−7 y+2 z=11 and containing the point (8,11 ,−3 ).

25. Parallel to the plane 5 x−3 y−z=−4 and containing the point (4 ,−6,1 ).

26. Perpendicular to n=⟨ 4,3 ,−2 ⟩ and having x-intercept 5.

In exercises 27 – 29, find a particular equation of the plane containing the given points.

27. (3,5,8 ) , (−2,4,1 ) , (−4,7,3 )

28. (5,7,3 ) , (4 ,−2,6 ) , (2 ,−6,1 )

29. (0,3 ,−7 ) , (5,0 ,−1 ) , (4,3,9 )

30. The cross product of the normal vectors to two planes is a vector that points in the direction of the line of intersection of the planes. Find a particular equation of the plane containing the point (−3,6,5 ) and normal to the line of intersection of the planes 3 x+5 y+4 z=−13 and 6 x−2 y+7 z=8.


10.5 Plane Curves and Parametric EquationsLet f and g be continuous functions of t on an interval I . The set of all points ( x , y ), where x=f (t ) and y=g (t ) is called a plane curve. The variable t is called a parameter, and the equations that define x and y are called parametric equations. A pair of parametric equations that describe a given curve is called a parameterization of the curve. More than one parameterization is possible for a given curve.

Example 1 – Find three parameterizations of the line through (1 ,−3 ) with slope −2.

Example 2 – By hand, graph the curve given by x=−2 t and y=4 t 2−4 , −1≤t ≤2.

Example 3 – Consider the curve given by x=−2 t and y=4 t 2−4 , −1≤t ≤2. Find an equation in x and y whose graph includes the graph of the given curve.

Example 4 – Eliminate the parameter in the equations x=5cos t+4 , y=2sin t−3, 0≤ t ≤2π .

Holt – Sections 11.7 & 11.7a

10.5 Plane Curves and Parametric Equations

Example 5 – Convert ( y−5 )2

9−

( x+2 )2

16=1 into parametric equations.

Example 6 – Given the parent relation x= y2, write a set of parametric equations to represent the relation, and sketch the graph. Then write parametric equations of the following successive transformations of the parent relation, and sketch each graph: (a) a horizontal dilation by a factor of 5; (b) then a horizontal shift 3 units to the right; (c) then a vertical shift down 2 units.

When a projectile

Is fired from the position (0 , k ) on the positive y-axis at an angle of θ with the horizontal, In the direction of the positive x-axis, With initial velocity v feet per second, With negligible air resistance,

Then its position at time t seconds is given by the parametric equations x=( vcosθ )t and y= (v sinθ )t+k−16 t2. If the initial velocity is given as v meters per second, the y equation becomes

y= (v sinθ )t+k−4.9 t 2.

Example 5 – A golfer hits a ball with an initial velocity of 140 feet per second so that its path as it leaves the ground makes an angle of 31° with the horizontal. (a) When does the ball hit the ground? (b) How far from its starting point does it land? (c) What is the maximum height of the ball during its flight?


10.5 Plane Curves and Parametric EquationsExample 6 – A batter hits a ball that is 3 feet above the ground. The ball leaves the bat with an initial velocity of 138 feet per second, making an angle of 26° with the horizontal and heading toward a 25-foot fence that is 400 feet away. Will the ball go over the fence?


In exercises 1 – 6, the given curve is part of the graph of an equation in x and y. Eliminate the parameter.

1. x=t+5 , y=2 t+1 , t ≥0

2. x=t+5 , y=√ t , t ≥0

3. x=t 2+1, y=t2−1 , for any t

4. x=−2+t 2 , y=1+2t 2, for any t

5. x=4sin 2t , y=2cos2 t ,0≤t ≤2π

6. x=2sin t−3 , y=2cos t+1 ,0≤ t ≤2 π

In exercises 7 – 16, find parametric equations for the curve whose equation is given.

7. y2

49+ x2

81=1

8. x2+4 y2=1

9. y2

9− x2

16=1

10. 2 x2− y2=4

11.( x−2 )2

16+

( y+3 )2

12=1

12.( x+5 )2

4+

( y+2 )2

12=1

13. y=3 ( x−2 )2−3

14. x=−3 ( y−1 )2−2

15.( y+1 )2

9−

( x−1 )2

25=1

16.( y+5 )2

9−

( x−2 )2

1=1

In exercises 17 – 18, sketch the graphs of the given curves and compare them. Do they differ? If so, how?

17.

a. x=−4+6 t , y=7−12 t ,0≤t ≤1

b. x=2−6 t , y=−5+12 t ,0≤ t ≤1

18.

a. x=t , y=t2, for any t

b. x=√ t , y=t, for any t

c. x=e t , y=e2 t , for any t

In exercises 19 – 20, find a parameterization of the given curve.


10.5 Plane Curves and Parametric Equations19. Line segment from (14 ,−5 ) to (5 ,−14 ) 20. Line segment from (18,4 ) to (−16,14 )


10.5 Plane Curves and Parametric EquationsIn exercises 21 – 26, assume that air resistance is negligible.

21. A ball is thrown from a height of 5 feet above the ground with an initial velocity of 60 feet per second at an angle of 50° with the horizontal. When and where does the ball hit the ground?

22. A medieval bowman shoots an arrow which leaves the bow 4 feet above the ground with an initial velocity of 88 feet per second with an angle of elevation of 48°. Will the arrow go over the 40-foot castle wall that is 200 feet from the archer?

23. A golfer at a driving range stands on a platform 2 feet above the ground and hits the ball with an initial velocity of 120 feet per second at an angle of elevation of 39°. There is a 32-foot-high fence 400 feet away. Will the ball fall short, hit the fence, or go over it?

24. A golf ball is hit off the tee at an angle of 30° and lands 300 feet away. What was its initial velocity?

25. A football kicked from the ground has an initial velocity of 75 feet per second.

a. Find the angle needed for the ball to travel exactly 150 feet.

b. Set up the parametric equations that describe the ball’s path.

26. A skeet is fired from the ground with an initial velocity of 110 feet per second at an angle of 28°.

a. How long is the skeet in the air?

b. How high does it go?


Module 10 – Selected Solutions

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· Web viewAn Airbus A320 jet maintains a constant airspeed of 500 mph headed due north. The jet stream is 100 mph in the northeasterly direction (N45 E). Express the velocity v