1.
The waterfall is 3 km away in a direction [latex]15°[/latex] south of west.
Chapter 9: Vectors
Exercises: 9.1 Geometric Form
Skills
- Sketch a vector #1–6
- Identify equal vectors #7–10
- Sketch a scalar multiple of a vector #11–14
- Sketch the sum of two vectors #15–22
- Calculate a resultant vector #23–32, 45–48
- Use vectors to solve problems #33–36
- Find components of a vector #37–40
- Find the magnitude and direction of a vector given in components #41–44
- Subtract vectors #49–58
Suggested Homework Problems
Problems: #3, 7, 13, 15, 23, 29, 35, 37, 41, 47
Homework 9-1
Exercise Group
For Problems 1–6, sketch a vector to represent the quantity.
2.
The cave entrance is 450 meters away, [latex]45°[/latex] north of east.
3.
The current is moving 6 feet per second in a direction [latex]60°[/latex] east of north.
4.
The bird is flying due south at 45 miles per hour.
5.
The projectile was launched at a speed of 40 meters per second at an angle of [latex]30°[/latex] above horizontal.
6.
The baseball was hit straight up at a speed of 60 miles per hour.
Exercise Group
For Problems 7–10, which vectors are equal?
7.
8.
9.
10.
Exercise Group
For Problems 11–14, sketch a vector equal to [latex]\bf{v}\text{,}[/latex] but starting at the given point.
11.
[latex](4,-1)[/latex]
12.
[latex](-3,1)[/latex]
13.
[latex](0,-2)[/latex]
14.
[latex](-3,-1)[/latex]
Exercise Group
For Problems 15–18, draw the scalar multiples of the given vectors.
15.
[latex]-2\bf{v}[/latex] and [latex]1.5\bf{v}[/latex]
16.
[latex]\dfrac{-1}{2}\bf{w}[/latex] and [latex]3\bf{w}[/latex]
17.
[latex]-2.5\bf{u}[/latex] and [latex]\sqrt{2}\bf{u}[/latex]
18.
[latex]-\sqrt{6}\bf{t}[/latex] and [latex]5.4\bf{t}[/latex]
Exercise Group
For Problems 19–26,
- draw the resultant vector,
- calculate the length and direction of the resultant vector.
19.
[latex]\bf{A} = \bf{u} + \bf{v}[/latex]
20.
[latex]\bf{B} = \bf{z} + \bf{u}[/latex]
21.
[latex]\bf{C} = \bf{w} + \bf{u}[/latex]
22.
[latex]\bf{D} = \bf{G} + \bf{z}[/latex]
23.
[latex]\bf{E} = \bf{z} + \bf{F}[/latex]
24.
[latex]\bf{F} = \bf{w} + \bf{v}[/latex]
25.
[latex]\bf{G} = \bf{w} + \bf{w}[/latex]
26.
[latex]\bf{H} = \bf{G} + \bf{G}[/latex]
Exercise Group
For Problems 27–30, find the magnitude and direction of the vector.
27.
[latex]v_x = 5,~ v_y = -12[/latex]
28.
[latex]v_x = -8,~ v_y = 15[/latex]
29.
[latex]v_x = -6,~ v_y = -7[/latex]
30.
[latex]v_x = 1,~ v_y = -3[/latex]
Exercise Group
For Problems 31–38, sketch the vectors, then calculate the resultant.
31.
Add the vector [latex]\bf{v}[/latex] of length 45 pointing [latex]26°[/latex] east of north to the vector [latex]\bf{w}[/latex] of length 32 pointing [latex]17°[/latex] south of west.
32.
Add the vector [latex]\bf{v}[/latex] of length 105 pointing [latex]41°[/latex] west of south to the vector [latex]\bf{w}[/latex] of length 77 pointing [latex]8°[/latex] west of north.
33.
Let [latex]\bf{v}[/latex] have length 8 and point in the direction [latex]80°[/latex] counterclockwise from the positive [latex]x[/latex]-axis. Let [latex]\bf{w}[/latex] have length 13 and point in the direction [latex]200°[/latex] counterclockwise from the positive [latex]x[/latex]-axis. Find [latex]\bf{v}+\bf{w}\text{.}[/latex]
34.
Let [latex]\bf{a}[/latex] have length 43 and point in the direction [latex]107°[/latex] counterclockwise from the positive [latex]x[/latex]-axis. Let [latex]\bf{b}[/latex] have length 19 and point in the direction [latex]309°[/latex] counterclockwise from the positive [latex]x[/latex]-axis. Find [latex]\bf{a}+\bf{b}\text{.}[/latex]
35.
Esther swam 3.6 miles heading [latex]20°[/latex] east of north. However, the water current displaced her by 0.9 miles in the direction [latex]37°[/latex] east of north. How far is Esther from her starting point, and in what direction?
36.
Rani paddles her canoe 4.5 miles in the direction [latex]12°[/latex] west of north. The water current pushes her 0.3 miles off course in the direction [latex]5°[/latex] east of north. How far is Rani from her starting point, and in what direction?
37.
Brenda wants to fly to an airport that is 103 miles due west in 1 hour. The prevailing winds blow in the direction [latex]112°[/latex] east of north at 28 miles per hour, so Brenda will head her plane somewhat north of due west to compensate. What airspeed and direction should Brenda take?
38.
Ryan wants to cross a 300-meter-wide river running due south at 80 meters per minute. There are rocks upstream and rapids downstream, so he wants to paddle straight across from east to west. In what direction should he point his kayak, and how fast should his water speed be in order to cross the river in 2 minutes? (Hint: The current will move him 160 meters due south compared with where his speed and direction would take him if the current stopped. Compute the distance he would have traveled, then divide by 2 minutes to get the speed.)
Exercise Group
For Problems 39–42,
- find the horizontal and vertical components of the vectors,
- use the components to calculate the resultant vector.
39.
A ship maintains a heading of [latex]30°[/latex] east of north and a speed of 20 miles per hour. There is a current in the water running [latex]45°[/latex] south of east at a speed of 10 miles per hour. What is the actual direction and speed of the ship?
40.
A plane is heading due south with an airspeed of 180 kilometers per hour. The wind is blowing at 50 kilometers per hour in a direction [latex]45°[/latex] south of west. What is the actual direction and speed of the plane?
41.
The campground is 3.6 kilometers from the trailhead in the direction [latex]20°[/latex] west of north. A ranger station is located 2.3 kilometers from the campsite in a direction of [latex]8°[/latex] west of south. What is the distance and direction from the trailhead to the ranger station?
42.
The treasure is buried 40 paces due east from the dead tree. From the buried treasure, a hidden mine shaft is 100 paces distant in a direction of [latex]32°[/latex] north of west. What is the distance and direction from the dead tree to mine shaft?
Exercise Group
Subtracting Vectors
Multiplying a vector [latex]\bf{v}[/latex] by [latex]-1[/latex] gives a vector [latex]-\bf{v}[/latex] that has the same magnitude as [latex]\bf{v}[/latex] but points in the opposite direction. We define subtraction of two vectors the same way we define subtraction of integers:
[latex]\bf{u} - \bf{v} = \bf{u} + (-\bf{v})[/latex]
That is, to subtract a vector [latex]\bf{v}\text{,}[/latex] we add its opposite.
For Problems 43–50, draw the resultant vector.
43.
[latex]\bf{A} = \bf{u} - \bf{v}[/latex]
44.
[latex]\bf{B} = \bf{F} - \bf{z}[/latex]
45.
[latex]\bf{C} = \bf{v} - \bf{u}[/latex]
46.
[latex]\bf{D} = \bf{z} - \bf{G}[/latex]
47.
[latex]\bf{P} = \bf{w} - \bf{F}[/latex]
48.
[latex]\bf{Q} = \bf{u} - \bf{w}[/latex]
49.
[latex]\bf{R} = \bf{G} - \bf{u}[/latex]
50.
[latex]\bf{S} = \bf{v} - \bf{F}[/latex]
51.
Find the horizontal and vertical components of [latex]\bf{u}\text{,}[/latex] [latex]\bf{v}\text{,}[/latex] and [latex]\bf{A}[/latex] from Problem 43. What do you notice when you compare the horizontal components of two vectors with the horizontal component of the difference?
52.
Find the horizontal and vertical components of [latex]\bf{z}\text{,}[/latex] [latex]\bf{y}\text{,}[/latex] and [latex]\bf{B}[/latex] from Problem 44. What do you notice when you compare the vertical components of two vectors with the vertical component of the difference?