Download Selected Solutions Pre-AP Precalculus Module 10

Pre-AP Precalculus
Module 10
Vectors and Parametric
Equations
10.1 Vectors
A 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
⃗⃗⃗⃗⃗ has initial point 𝑃 and terminal point 𝑄. If vector 𝒗 has the same magnitude as vector 𝑃𝑄
⃗⃗⃗⃗⃗ , then
vector 𝑃𝑄
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
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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 𝛼𝒗 is defined as:
o If 𝛼 > 0, 𝛼𝒗 is the vector whose magnitude is 𝛼 times the magnitude of 𝒗 and whose direction is the
same as 𝒗.
o If 𝛼 < 0, 𝛼𝒗 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 𝛼𝒗 = 𝟎.
(𝛼 + 𝛽)𝒗 = 𝛼𝒗 + 𝛽𝒗
𝛼(𝒗 + 𝒘) = 𝛼𝒗 + 𝛼𝒘
𝛼(𝛽𝒗) = (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 Vectors
We use the symbol ‖𝒗‖ to represent the magnitude of a vector 𝒗. Since ‖𝒗‖ equals the length of a
directed line segment,
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‖𝒗‖ ≥ 0
‖𝒗‖ = 0 if and only if 𝒗 = 𝟎
‖𝒗‖ = ‖−𝒗‖
‖𝛼𝒗‖ = |𝛼|‖𝒗‖
A vector 𝒖 for which ‖𝒖‖ = 1 is called a unit vector, and is denoted as 𝑢̂.
𝑖̂ = 〈1,0〉 is the unit vector in the direction of the positive 𝑥-axis, and 𝑗̂ = 〈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
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𝑎𝑖̂ is a vector with magnitude 𝑎 in the direction of 𝑖̂
𝑏𝑗̂ is the vector with magnitude 𝑏 in the direction of 𝑗̂
−𝑎𝑖̂ is the vector with magnitude 𝑎 in the opposite direction of 𝑖̂
−𝑏𝑗̂ is the vector with magnitude 𝑏 in the direction of 𝑗̂
𝒗 = 𝑎𝑖̂ + 𝑏𝑗̂ = 〈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 = ⃗⃗⃗⃗⃗⃗⃗⃗
𝑃1 𝑃2 , then 𝒗 is equal to the position vector 〈𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 〉.
Example 2 – Find the displacement vector ⃗⃗⃗⃗⃗⃗⃗⃗
𝑃1 𝑃2 if 𝑃1 = (−1,2) and 𝑃2 = (4,6).
If 𝒗 = 〈𝑎1 , 𝑏1 〉 and 𝒘 = 〈𝑎2 , 𝑏2 〉, then:
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𝒗 = 𝒘 if and only if 𝑎1 = 𝑎2 and 𝑏1 = 𝑏2 .
𝒗 ± 𝒘 = 〈𝑎1 ± 𝑎2 , 𝑏1 ± 𝑏2 〉 = (𝑎1 ± 𝑎2 )𝑖̂ + (𝑏1 ± 𝑏2 )𝑗̂
𝛼𝒗 = 〈𝛼𝑎1 , 𝛼𝑏1 〉 = 𝛼𝑎1 𝑖̂ + 𝛼𝑏1 𝑗̂
Sullivan & Sullivan – Section 9.4
10.1 Vectors
Just like with polar complex notation, if 𝒗 = 〈𝑎, 𝑏〉, then ‖𝒗‖ = √𝑎2 + 𝑏 2 and the direction of 𝒗 is 𝜃𝑣 =
𝑏
𝑏
tan−1 𝑎 or 𝜃𝑣 = tan−1 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 =
〈𝑎 cos 𝜃𝑤 , 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‖.
𝒗
For any nonzero vector 𝒗, the vector 𝑣̂ = ‖𝒗‖ is a unit vector that has the same direction as 𝒗. As a
consequence, 𝒗 = ‖𝒗‖𝑣̂.
Example 5 – Find a unit vector in the same direction as 𝒗 = 〈4, −3〉.
Sullivan & Sullivan – Section 9.4
10.1 Vectors
Example 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 𝑖̂ and 𝑗̂. 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 𝑖̂ and 𝑗̂. (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.
Sullivan & Sullivan – Section 9.4
10.1 Vectors
Example 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
B
A
F
G
H
3. 𝑪 = 𝑫 − 𝑬 + 𝑭
E
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
Sullivan & Sullivan – Section 9.4
C
K
D
10.1 Vectors
In exercises 9 – 12, find the displacement vector ⃗⃗⃗⃗⃗⃗
𝑷𝑸 in the form 〈𝒂, 𝒃〉.
9. 𝑃 = (0,0); 𝑄 = (−3, −5)
11. 𝑃 = (−1,4); 𝑄 = (6,2)
10. 𝑃 = (−3,2); 𝑄 = (6,5)
12. 𝑃 = (1,1); 𝑄 = (2,2)
In exercises 13 – 18, find ‖𝒗‖.
13. 𝒗 = 3𝑖̂ − 4𝑗̂
16. 𝒗 = −𝑖̂ − 𝑗̂̂
14. 𝒗 = −5𝑖̂ + 12𝑗̂
17. 𝒗 = −2𝑖̂ + 3𝑗̂
15. 𝒗 = 𝑖̂ − 𝑗̂̂
18. 𝒗 = 6𝑖̂ + 2𝑗̂̂
In exercises 19 – 24, find each quantity if 𝒗 = 𝟑𝒊̂ − 𝟓𝒋̂ and 𝒘 = −𝟐𝒊̂ + 𝟑𝒋̂.
19. 2𝒗 + 3𝒘
22. ‖𝒗 + 𝒘‖
20. 3𝒗 − 2𝒘
23. ‖𝒗‖ − ‖𝒘‖
21. ‖𝒗 − 𝒘‖
24. ‖𝒗‖ + ‖𝒘‖
In exercises 25 – 28, find the unit vector in the same direction as 𝒗.
25. 𝒗 = 3𝑖̂ − 4𝑗̂
27. 𝒗 = 𝑖̂ − 𝑗̂̂
26. 𝒗 = −5𝑖̂ + 12𝑗̂
28. 𝒗 = 2𝑖̂ − 𝑗̂̂
29. Find a vector v whose magnitude is 4 and whose component in the 𝑖̂ direction is twice the component in the
𝑗̂ direction.
30. Find a vector v whose magnitude is 3 and whose component in the 𝑖̂ direction is equal to the component in
the 𝑗̂ direction.
31. If 𝒗 = 2𝑖̂ − 𝑗̂ and 𝒘 = 𝑥𝑖̂ + 3𝑗̂, find all numbers 𝑥 for which ‖𝒗 + 𝒘‖ = 5.
⃗⃗⃗⃗⃗ has length 5.
32. If 𝑃 = (−3,1) and 𝑄 = (𝑥, 4), find all numbers 𝑥 such that the vector represented by 𝑃𝑄
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°
35. ‖𝒗‖ = 3; 𝛼 = 240°
34. ‖𝒗‖ = 14; 𝛼 = 120°
36. ‖𝒗‖ = 15; 𝛼 = 315°
In exercises 37 – 40, find the magnitude and direction angle of 𝒗 for each vector.
37. 𝒗 = 𝑖̂ + √3𝑗̂̂
39. 𝒗 = 6𝑖̂ − 4𝑗̂̂
38. 𝒗 = −5𝑖̂ − 5𝑗̂̂
40. 𝒗 = −𝑖̂ + 3𝑗̂̂
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 .
Sullivan & Sullivan – Section 9.4
10.1 Vectors
42. 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 𝑖̂ and 𝑗̂.
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.
Sullivan & Sullivan – Section 9.4
10.2 The Dot Product
If 𝑣 = 𝑎1 𝑖̂ + 𝑏1 𝑗̂ and 𝑤
⃗⃗ = 𝑎2 𝑖̂ + 𝑏2 𝑗̂, are two vectors, then the dot or scalar product 𝑣 ∙ 𝑤
⃗⃗ = 𝑎1 𝑎2 + 𝑏1 𝑏2 .
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
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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‖ cos 𝜃. 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 = 4𝑖̂ − 3𝑗̂ and 𝒗 = 2𝑖̂ + 5𝑗̂.
Example 3 – Find the angle 𝜃 between 𝒗 = 〈3, −1〉 and 𝒘 = 〈6, −2〉.
Sullivan & Sullivan – Section 9.5
Foerster – Section 10.4
10.2 The Dot Product
Two 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‖ cos 𝜃. 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
𝒘
vector in the direction of w, namely ‖𝒘‖. Thus the vector projection 𝒑 =
‖𝒘‖
𝒘‖𝒗‖ cos 𝜃
.
‖𝒘‖
The numerator almost
𝒗∙𝒘
looks like 𝒗 ∙ 𝒘 = ‖𝒗‖‖𝒘‖ cos 𝜃, and by multiplying by ‖𝒘‖, we get the equivalent formula 𝒑 = ‖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.
Sullivan & Sullivan – Section 9.5
Foerster – Section 10.4
10.2 The Dot Product
Example 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
⃗⃗⃗⃗⃗ ‖. This definition presupposes
point 𝐵 is defined as 𝑊 = (𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑓𝑜𝑟𝑐𝑒)(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒) = ‖𝑭‖‖𝐴𝐵
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?
Complete the following exercises on a separate sheet of paper.
⃗ ∙𝒘
⃗ and 𝒘
In exercises 1 – 16, (a) find the dot product 𝒗
⃗⃗⃗ ; (b) find the angle between 𝒗
⃗⃗⃗ ; and (c) state
whether the vectors are parallel, orthogonal, or neither.
1. 𝑣 = 〈1, −1〉, 𝑤
⃗⃗ = 〈1,1〉
6. 𝑣 = 〈1, √3〉, 𝑤
⃗⃗ = 〈1, −1〉
2. 𝑣 = 〈1,1〉, 𝑤
⃗⃗ = 〈−1,1〉
7. 𝑣 = 〈3,4〉, 𝑤
⃗⃗ = 〈−6, −8〉
3. 𝑣 = 〈2,1〉, 𝑤
⃗⃗ = 〈1, −2〉
8. 𝑣 = 〈3, −4〉, 𝑤
⃗⃗ = 〈9, −12〉
4. 𝑣 = 〈2,2〉, 𝑤
⃗⃗ = 〈1,2〉
9. 𝑣 = 4𝑖̂, 𝑤
⃗⃗ = 𝑗̂
5. 𝑣 = 〈√3, −1〉, 𝑤
⃗⃗ = 〈1,1〉
10. 𝑣 = 𝑖̂, 𝑤
⃗⃗ = −3𝑗̂
11. Find 𝑎 so that the vectors 〈1, −𝑎〉 and 〈2,3〉 are orthogonal.
Sullivan & Sullivan – Section 9.5
Foerster – Section 10.4
10.2 The Dot Product
12. Find 𝑏 so that the vectors 〈1,1〉 and 〈1, 𝑏〉 are orthogonal.
⃗ into two vectors ⃗⃗⃗⃗
In exercises 13 – 18, decompose 𝒗
𝒗𝟏 and ⃗⃗⃗⃗
𝒗𝟐 , where ⃗⃗⃗⃗
𝒗𝟏 is parallel to 𝒘
⃗⃗⃗ and ⃗⃗⃗⃗
𝒗𝟐 is
orthogonal to 𝒘
⃗⃗⃗ .
13. 𝑣 = 〈2, −3〉, 𝑤
⃗⃗ = 〈1, −1〉
16. 𝑣 = 〈2, −1〉, 𝑤
⃗⃗ = 〈1, −2〉
14. 𝑣 = 〈−3,2〉, 𝑤
⃗⃗ = 〈2,1〉
17. 𝑣 = 〈3,1〉, 𝑤
⃗⃗ = 〈−2, −1〉
15. 𝑣 = 〈1, −1〉, 𝑤
⃗⃗ = 〈−1, −2〉
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
place 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?
Sullivan & Sullivan – Section 9.5
Foerster – Section 10.4
10.2 The Dot Product
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.
Sullivan & Sullivan – Section 9.5
Foerster – Section 10.4
10.3 Vectors in Space
Coordinates 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 𝑖̂, 𝑗̂ and 𝑘̂, whose directions are along the positive 𝑥axis, 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 𝑎𝑖̂ + 𝑏𝑗̂ + 𝑐𝑘̂ = 〈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 ⃗⃗⃗⃗⃗⃗⃗⃗
𝑃1 𝑃2 =
̂
(𝑥2 − 𝑥1 )𝑖̂ + (𝑦2 − 𝑦1 )𝑗̂ + (𝑧2 − 𝑧1 )𝑘.
Example 2 – Find the displacement vector ⃗⃗⃗⃗⃗⃗⃗⃗
𝑃1 𝑃2 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 )𝑖̂ + (𝑏1 ± 𝑏2 )𝑗̂ + (𝑐1 + 𝑐2 )𝑘̂
𝛼𝑣 = (𝛼𝑎1 )𝑖̂ + (𝛼𝑏1 )𝑗̂ + (𝛼𝑐1 )𝑘̂
‖𝑣 ‖ = √𝑎12 + 𝑏12 + 𝑐12
Sullivan & Sullivan – Section 9.6
10.3 Vectors in Space
Example 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) 3𝑣, (b) 2𝑣 − 3𝑤
⃗⃗ , and (c) ‖𝑣‖.
⃗
𝑣
Recall 𝑣̂ = ‖𝑣⃗‖ is a unit vector in the direction of 𝑣, and as a consequence 𝑣 = ‖𝑣‖𝑣̂.
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
⃗⃗ = 𝑎1 𝑎2 + 𝑏1 𝑏2 + 𝑐1 𝑐2 =
‖𝑣‖‖𝑤
⃗⃗ ‖ cos 𝜃.
Sullivan & Sullivan – Section 9.6
10.3 Vectors in Space
Example 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
⃗⃗
2
𝑣 ∙ 𝑣 = ‖𝑣 ‖
⃗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 𝑖̂, 0 ≤ 𝛼 ≤ 𝜋
𝛽 = the angle between 𝑣 and 𝑗̂, 0 ≤ 𝛽 ≤ 𝜋
𝛾 = the angle between 𝑣 and 𝑘̂, 0 ≤ 𝛾 ≤ 𝜋
𝑎
𝑏
If 𝑣 = 〈𝑎, 𝑏, 𝑐〉 is a nonzero vector in space, the direction angles 𝛼, 𝛽, and 𝛾 obey cos 𝛼 = ‖𝑣⃗‖, cos 𝛽 = ‖𝑣⃗‖,
𝑐
and cos 𝛾 = ‖𝑣⃗‖.
Example 8 – Find the direction angles of 〈−3,2, −6〉.
Sullivan & Sullivan – Section 9.6
10.3 Vectors in Space
If 𝛼, 𝛽, and 𝛾 are the direction angles of a nonzero vector in space, then cos2 𝛼 + cos 2 𝛽 + cos 2 𝛾 = 1.
𝜋
𝜋
Example 9 – The vector 𝑣 makes an angle of 𝛼 = 3 with the positive 𝑥-axis, an angle of 𝛽 = 3 with the
positive 𝑦-axis, and an acute angle 𝛾 with the positive 𝑧-axis. Find 𝛾.
It can be shown that 𝑣 = ‖𝑣 ‖[(cos 𝛼)𝑖̂ + (cos 𝛽)𝑗̂ + (cos 𝛾)𝑘̂].
2
Example 10 – Find the point that is 3 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〉.
Sullivan & Sullivan – Section 9.6
10.3 Vectors in Space
Complete 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)
3. 𝑃 = (−3,2,0) and 𝑄 = (6,5, −1)
2. 𝑃 = (3,2, −1) and 𝑄 = (5,6,0)
4. 𝑃 = (−1,4, −2) and 𝑄 = (6,2,2)
⃗ ‖.
In exercises 5 – 8, find ‖𝒗
5. 𝑣 = 〈3, −6, −2〉
7. 𝑣 = 〈−1, −1,1〉
6. 𝑣 = 〈−6,12,4〉
8. 𝑣 = 〈6,2, −2〉
̂ and 𝒘
̂.
⃗ = 𝟑𝒊̂ − 𝟓𝒋̂ + 𝟐𝒌
In exercises 9 – 14, find each requested quantity if 𝒗
⃗⃗⃗ = −𝟐𝒊̂ + 𝟑𝒋̂ − 𝟐𝒌
9. 2𝑣 + 3𝑤
⃗⃗
12. ‖𝑣 + 𝑤
⃗⃗ ‖
10. 3𝑣 − 2𝑤
⃗⃗
13. ‖𝑣 ‖ − ‖𝑤
⃗⃗ ‖
11. ‖𝑣 − 𝑤
⃗⃗ ‖
14. ‖𝑣 ‖ + ‖𝑤
⃗⃗ ‖
̂ in the same direction as 𝒗
⃗.
In exercises 15 – 20, find the unit vector 𝒗
15. 𝑣 = 5𝑖̂
18. 𝑣 = −6𝑖̂ + 12𝑗̂ + 4𝑘̂
16. 𝑣 = −2𝑗̂
19. 𝑣 = 𝑖̂ + 𝑗̂ + 𝑘̂
17. 𝑣 = 3𝑖̂ − 6𝑗̂ − 2𝑘̂
20. 𝑣 = 2𝑖̂ − 𝑗̂ + 𝑘̂
⃗ ∙𝒘
⃗ and 𝒘
In exercises 21 – 24, find the dot product 𝒗
⃗⃗⃗ and the angle between 𝒗
⃗⃗⃗ .
21. 𝑣 = 𝑖̂ + 𝑗̂, 𝑤
⃗⃗ = −𝑖̂ + 𝑗̂ − 𝑘̂
23. 𝑣 = 𝑖̂ + 3𝑗̂ + 2𝑘̂ , 𝑤
⃗⃗ = 𝑖̂ − 𝑗̂ + 𝑘̂
22. 𝑣 = 2𝑖̂ + 2𝑗̂ − 𝑘̂ , 𝑤
⃗⃗ = 𝑖̂ + 2𝑗̂ + 3𝑘̂
24. 𝑣 = 3𝑖̂ − 4𝑗̂ + 𝑘̂ , 𝑤
⃗⃗ = 6𝑖̂ − 8𝑗̂ + 2𝑘̂
⃗ =
In exercises 25 – 28, find the direction angles of each vector. Write each vector in the form 𝒗
̂
‖𝒗
⃗ ‖[(𝒄𝒐𝒔 𝜶)𝒊̂ + (𝒄𝒐𝒔 𝜷)𝒋̂ + (𝒄𝒐𝒔 𝜸)𝒌].
25. 𝑣 = 〈−6,12,4〉
27. 𝑣 = 〈0,1,1〉
26. 𝑣 = 〈1, −1, −1〉
28. 𝑣 = 〈2,3, −4〉
In exercises 29 – 32, find the radius and center of each sphere.
29. 𝑥 2 + 𝑦 2 + 𝑧 2 + 2𝑥 − 2𝑧 = 2
31. 𝑥 2 + 𝑦 2 + 𝑧 2 − 4𝑥 = 0
30. 𝑥 2 + 𝑦 2 + 𝑧 2 − 4𝑥 + 4𝑦 + 2𝑧 = 0
32. 3𝑥 2 + 3𝑦 2 + 3𝑧 2 + 6𝑥 − 6𝑦 = 3
33. Find the work done by a force of 3 newtons acting in the direction 2𝑖̂ + 𝑗̂ + 2𝑘̂ 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
⃗ = 3𝑖̂ + 2𝑗̂ − 5𝑘̂ if the applied force is 𝐹̂ = 2𝑖̂ − 𝑗̂ −
𝑘̂. Use meters for distance and newtons for force.
35. Find the point that is 75% of the way from (−3,5,2) to (1, −3,14).
Sullivan & Sullivan – Section 9.6
10.3 Vectors in Space
1
36. Find the point that is 3 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〉.
Sullivan & Sullivan – Section 9.6
10.4 The Cross Product and Planar Equations
If 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 )𝑖̂ − (𝑎1 𝑐2 − 𝑎2 𝑐1 )𝑗̂ + (𝑎1 𝑏2 − 𝑎2 𝑏1 )𝑘̂.
Example 1 – If 𝑣 = 〈2,3,5〉 and 𝑤
⃗⃗ = 〈1,2,3〉, find 𝑣 × 𝑤
⃗⃗ .
𝐴
2 3
Example 2 – Calculate |
| and | 2
1 2
1
𝐵
3
2
𝐶
5 |.
3
𝑖̂
The cross product of the vectors 𝑣 = 〈𝑎1 , 𝑏1 , 𝑐1 〉 and 𝑤
⃗⃗ = 〈𝑎2 , 𝑏2 , 𝑐2 〉 is 𝑣 × 𝑤
⃗⃗ = |𝑎1
𝑎2
𝑗̂
𝑏1
𝑏2
𝑘̂
𝑐1 |.
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 9.7
Foerster – Sections 10.5 & 10.6
10.4 The Cross Product and Planar Equations
If 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 ) = (𝛼𝑢
⃗)×𝑣 =𝑢
⃗ × (𝛼𝑣 )
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 ‖ sin 𝜃 where 𝜃 is the angle between 𝑢
⃗ and 𝑣 when placed tail-to-tail.
⃗ and 𝑣 ≠ 0
⃗ as adjacent sides.
‖𝑢
⃗ × 𝑣 ‖ is the area of the parallelogram having 𝑢
⃗ ≠0
1
⃗ and 𝑣 ≠ 0
⃗ as adjacent sides.
‖𝑢
⃗ × 𝑣 ‖ is the area of the triangle having 𝑢
⃗ ≠0
2

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 𝑛⃗ = 〈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 𝑛⃗ = 〈11,2,13〉.
Sullivan & Sullivan – Section 9.7
Foerster – Sections 10.5 & 10.6
10.4 The Cross Product and Planar Equations
Example 7 – Find two vectors ⃗⃗⃗⃗
𝑛1 and ⃗⃗⃗⃗
𝑛2 normal to the plane 7𝑥 − 3𝑦 + 8𝑧 = −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).
Complete the following exercises on a separate sheet of paper.
In exercises 1 – 4, find ⃗𝒗 × 𝒘
⃗⃗⃗ and 𝒘
⃗⃗⃗ × ⃗𝒗.
1. 𝑣 = 〈−1,3,2〉, 𝑤
⃗⃗ = 〈3, −2, −1〉
3. 𝑣 = 〈3,1,3〉, 𝑤
⃗⃗ = 〈1,0, −1〉
2. 𝑣 = 〈1, −4,2〉, 𝑤
⃗⃗ = 〈3,2,1〉
4. 𝑣 = 〈2, −3,0〉, 𝑤
⃗⃗ = 〈0,3, −2〉
Sullivan & Sullivan – Section 9.7
Foerster – Sections 10.5 & 10.6
10.4 The Cross Product and Planar Equations
⃗ = 〈𝟐, −𝟑, 𝟏〉, 𝒗
⃗ = 〈−𝟑, 𝟑, 𝟐〉, and 𝒘
In exercises 5 – 14, find each expression given that 𝒖
⃗⃗⃗ = 〈𝟏, 𝟏, 𝟑〉.
5. 𝑣 × 𝑤
⃗⃗
10. (𝑢
⃗ × 𝑣) ∙ 𝑤
⃗⃗
6. 𝑤
⃗⃗ × 𝑢
⃗
11. 𝑣 ∙ (𝑢
⃗ ×𝑤
⃗⃗ )
7. 𝑣 × 4𝑤
⃗⃗
12. (𝑢
⃗ × 𝑣) × 𝑤
⃗⃗
8. −3𝑢
⃗ ×𝑤
⃗⃗
13. Find a vector orthogonal to both 𝑣 and 𝑖̂ + 𝑗̂.
9. 𝑣 ∙ (𝑣 × 𝑤
⃗⃗ )
14. Find a vector orthogonal to both 𝑢
⃗ and 𝑗̂ + 𝑘̂.
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. 3𝑥 + 5𝑦 − 7𝑧 = −13
20. 4𝑥 − 7𝑦 + 2𝑧 = 9
In exercises 21 -26, find a particular equation of the plane described.
21. Perpendicular to 𝑛⃗ = 〈3, −5,4〉, containing the point (6, −7, −2).
22. Perpendicular to 𝑛⃗ = 〈−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 3𝑥 − 7𝑦 + 2𝑧 = 11 and containing the point (8,11, −3).
25. Parallel to the plane 5𝑥 − 3𝑦 − 𝑧 = −4 and containing the point (4, −6,1).
26. Perpendicular to 𝑛⃗ = 〈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)
29. (0,3, −7), (5,0, −1), (4,3,9)
28. (5,7,3), (4, −2,6), (2, −6,1)
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 3𝑥 + 5𝑦 + 4𝑧 = −13 and 6𝑥 − 2𝑦 + 7𝑧 = 8.
Sullivan & Sullivan – Section 9.7
Foerster – Sections 10.5 & 10.6
10.5 Plane Curves and Parametric Equations
Let 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 = −2𝑡 and 𝑦 = 4𝑡 2 − 4, −1 ≤ 𝑡 ≤ 2.
Example 3 – Consider the curve given by 𝑥 = −2𝑡 and 𝑦 = 4𝑡 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 = 5 cos 𝑡 + 4, 𝑦 = 2 sin 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 = 𝑦 2, 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 = (𝑣 cos 𝜃)𝑡 and 𝑦 =
(𝑣 sin 𝜃)𝑡 + 𝑘 − 16𝑡 2 . If the initial velocity is given as 𝑣 meters per second, the 𝑦 equation
becomes 𝑦 = (𝑣 sin 𝜃)𝑡 + 𝑘 − 4.9𝑡 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?
Holt – Sections 11.7 & 11.7a
10.5 Plane Curves and Parametric Equations
Example 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?
Complete the following exercises on a separate sheet of paper.
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 = 2𝑡 + 1, 𝑡 ≥ 0
4. 𝑥 = −2 + 𝑡 2 , 𝑦 = 1 + 2𝑡 2 , for any 𝑡
2. 𝑥 = 𝑡 + 5, 𝑦 = √𝑡, 𝑡 ≥ 0
5. 𝑥 = 4 sin 2𝑡 , 𝑦 = 2 cos 2𝑡 , 0 ≤ 𝑡 ≤ 2𝜋
3. 𝑥 = 𝑡 2 + 1, 𝑦 = 𝑡 2 − 1, for any 𝑡
6. 𝑥 = 2 sin 𝑡 − 3, 𝑦 = 2 cos 𝑡 + 1, 0 ≤ 𝑡 ≤ 2𝜋
In exercises 7 – 16, find parametric equations for the curve whose equation is given.
7.
𝑦2
49
+
𝑥2
81
12.
=1
8. 𝑥 2 + 4𝑦 2 = 1
9.
𝑦2
9
16
+
(𝑦+3)2
12
+
(𝑦+2)2
12
=1
14. 𝑥 = −3(𝑦 − 1)2 − 2
𝑥2
− 16 = 1
(𝑥−2)2
4
13. 𝑦 = 3(𝑥 − 2)2 − 3
15.
10. 2𝑥 2 − 𝑦 2 = 4
11.
(𝑥+5)2
=1
16.
(𝑦+1)2
9
(𝑦+5)2
9
−
−
(𝑥−1)2
25
(𝑥−2)2
1
=1
=1
In exercises 17 – 18, sketch the graphs of the given curves and compare them. Do they differ? If
so, how?
17.
18.
a. 𝑥 = −4 + 6𝑡, 𝑦 = 7 − 12𝑡, 0 ≤ 𝑡 ≤ 1
a. 𝑥 = 𝑡, 𝑦 = 𝑡 2 , for any 𝑡
b. 𝑥 = 2 − 6𝑡, 𝑦 = −5 + 12𝑡, 0 ≤ 𝑡 ≤ 1
b. 𝑥 = √𝑡, 𝑦 = 𝑡, for any 𝑡
c. 𝑥 = 𝑒 𝑡 , 𝑦 = 𝑒 2𝑡 , for any 𝑡
In exercises 19 – 20, find a parameterization of the given curve.
19. Line segment from (14, −5) to (5, −14)
Holt – Sections 11.7 & 11.7a
20. Line segment from (18,4) to (−16,14)
10.5 Plane Curves and Parametric Equations
In 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?
Holt – Sections 11.7 & 11.7a
Module 10 – Selected Solutions