1. rectangular components
vector
initial point
terminal point
standard position
direction
magnitude
quadrant bearing
true bearing
parallel vectors
equivalent vectors
opposite vectors
resultant
triangle method
parallelogram method
zero vector
components
rectangular components
Vocabulary

2. Utilizing your book, what is the difference between a scalar and a vector?

3. Identify Vector Quantities
A. State whether a hockey puck shot northwest at 60 miles per hour is a vector quantity or a scalar quantity.
This quantity has a magnitude of 60 miles per hour and a direction of northwest. This directed distance is a vector quantity.
Answer: vector
Example 1

4. Identify Vector Quantities
B. State whether a tennis ball served at 110 miles per hour is a vector quantity or a scalar quantity.
This quantity has a magnitude of 110 miles per hour, but no direction is given. Speed is a scalar quantity.
Answer: scalar
Example 1

5. Identify Vector Quantities
C. State whether a sprinter running 100 meters north is a vector quantity or a scalar quantity.
This quantity has a magnitude of 100 meters and a direction of north. This directed distance is a vector quantity.
Answer: vector
Example 1

6. State whether a tow truck pulling a car due east with a force of 100 newtons is a vector quantity or a scalar quantity.
A. vector
B. scalar
Example 1

7. Represent a Vector Geometrically
A. Use a ruler and a protractor to draw an arrow diagram for v = 10 newtons of force at 30° to the horizontal. Include a scale on the diagram.
Using a scale of 1 cm : 5 N, draw and label a 10 ÷ 5 or 2-cm arrow in standard position at a 30° angle to the x-axis.
Sample Answer: V
Example 2

8. Represent a Vector Geometrically
B. Use a ruler and a protractor to draw an arrow diagram for z = 25 meters per second at a bearing of S70°W. Include a scale on the diagram.
Using a scale of 1 cm : 10 m/s, draw and label a 25 ÷ 10 or 2.5-cm arrow 70° west of south.
Sample Answer:
Example 2

9. Represent a Vector Geometrically
C. Use a ruler and a protractor to draw an arrow diagram for t = 10 miles per hour at a bearing of 025°. Include a scale on the diagram.
Using a scale of 1 cm : 10 mi/h, draw and label a 10 ÷ 10 or 1-cm arrow at an angle of 25° clockwise from the north.
Sample Answer:
Example 2

10. Use a ruler and protractor to draw an arrow diagram for n = 20 feet per second at a bearing of 080°. Include a scale on the diagram. A. B. C. D.
Example 2

11. Video

12. Key Concept 3

13. Find the Resultant of Two Vectors
HIKING While hiking in the woods, Shelly walks 2 kilometers N30°W from her camp, and then walks 2 kilometers directly east. How far and at what quadrant bearing is Shelly from her camp?
Let p = walking 2 kilometers N30°W and q = walking 2 kilometers due east. Draw a diagram to represent p and q using a scale of 1 cm : 1 km.
Example 3

14. Find the Resultant of Two Vectors
Use a ruler and a protractor to draw a 2-centimeter arrow 30° west of north to represent p and a 2-centimeter arrow due east to represent q.
Example 3

15. Method 1 Triangle Method
Find the Resultant of Two Vectors
Method 1 Triangle Method
Translate q so that its tail touches the tip of p. Then draw the resultant vector p + q as shown.
Example 3

16. Method 2 Parallelogram Method
Find the Resultant of Two Vectors
Method 2 Parallelogram Method
Translate q so that its tail touches the tail of p. Then complete the parallelogram and draw the diagonal, resultant p + q, as shown.
Example 3

17. Find the Resultant of Two Vectors
Both methods produce the same resultant vector p + q. Measure the length of p + q and then measure the angle this vector makes with the north-south line as shown.
The vector’s length of 2 centimeters represents 2 kilometers. Therefore, Shelly is 2 kilometers at a bearing of 30° east of north or N30°E from her starting position.
Example 3

18. Find the Resultant of Two Vectors
Answer: 2 km, N30°E
Example 3

19. A. about 349 feet at a bearing of N86°E
ORIENTEERING In an orienteering competition, Miguel walks N80E for 150 feet and then walks 200 feet due east. How far and at what quadrant bearing is Miguel from his starting position?
A. about 349 feet at a bearing of N86°E
B. about 270 feet at a bearing of N37°E
C. about 270 feet at a bearing of N47°E
D. about 350 feet at a bearing of N80°E
Example 3

20. Key Concept 4

21. Draw a vector diagram of a – 3b.
Operations with Vectors
Draw a vector diagram of a – 3b.
Rewrite the expression as the addition of two vectors: a – 3b = a + (– 3b). To represent –3b, draw a vector 3 times as long as b in the opposite direction from b.
Example 4

22. Then use the triangle method to draw the resultant vector.
Operations with Vectors
Then use the triangle method to draw the resultant vector.
Answer:
Example 4

23. Draw a vector diagram of n + 2m.B. C. D.
Example 4

24. Step 1 Draw a diagram to represent the heading and wind velocities.
Use Vectors to Solve Navigation Problems
AVIATION An airplane is flying with an airspeed of 475 miles per hour on a heading of 070°. If an 80-mile-per-hour wind is blowing from a true heading of 120°, determine the velocity and direction of the plane relative to the ground.
Step 1 Draw a diagram to represent the heading and wind velocities.
Example 5

25. Use Vectors to Solve Navigation Problems
Translate the wind vector as shown in the diagram below, and use the triangle method to obtain the resultant vector representing the plane’s ground velocity and direction g. In the triangle formed by these vectors, γ = 120° – 70° or 50°.
Example 5

26. c 2 = a 2 + b 2 – 2ab cos γ Law of Cosines
Use Vectors to Solve Navigation Problems
Step 2 Use the Law of Cosines to find |g|, the plane’s speed relative to the ground.
c 2 = a 2 + b 2 – 2ab cos γ Law of Cosines
|g|2 = – 2(80)(475) cos 50° c = |g|, a = 80, b = 475, and γ = 50°
Take the positive square root of each side.
Example 5

27. The ground speed of the plane is about 428.0 miles per hour.
Use Vectors to Solve Navigation Problems
≈ Simplify.
The ground speed of the plane is about miles per hour.
Step 3 The heading of the resultant g is represented by angle θ, as shown above. To find θ, first calculate α using the Law of Sines.
Law of Sines
c = |g| or 428.0, a = 80, and γ = 50
Example 5

28. Apply the inverse sine function.
Use Vectors to Solve Navigation Problems
Solve for sin α.
Apply the inverse sine function.
Simplify.
The measure of θ is 70° – α, which is 70° – 8.2° or 61.8°.
Therefore, the speed of the plane relative to the ground is about miles per hour at about 061.8°.
Example 5

29. Use Vectors to Solve Navigation Problems
Answer: The velocity of the plane relative to the ground is about miles per hour at a bearing of about 061.8°.
Example 5

30. ROWING Jamie rows her boat due east at a speed of 20 feet per second across a river directly toward the opposite bank. At the same time, the current of the river is carrying her due south at a rate of 4 feet per second. Find Jamie’s speed and direction relative to the shore.
A. Jamie is rowing at a resultant velocity of 24 ft/s in a direction directly east.
B. Jamie is rowing at a resultant velocity of 20.4 ft/s in a direction of S11.3°E.
C. Jamie is rowing at a resultant velocity of 20.4 ft/s in a direction of S29.8°E.
D. Jamie is rowing at a resultant velocity of 20.4 ft/s in a direction of S78.7°E.
Example 5

31. Resolve a Force into Rectangular Components
A. GARDENING While digging in his garden, Will pushes a shovel into the ground with a force of 630 newtons at an angle of 70° with the ground. Draw a diagram that shows the resolution of the force that Will exerts into its rectangular components.
Example 6

32. Resolve a Force into Rectangular Components
Will’s push can be resolved into a horizontal push x forward and a vertical push y downward as shown.
Answer:
Example 6

33. Resolve a Force into Rectangular Components
B. GARDENING While digging in his garden, Will pushes a shovel into the ground with a force of 630 newtons at an angle of 70° with the ground. Find the magnitudes of the horizontal and vertical components of the force.
The horizontal and vertical components of the force form a right triangle. Use the sine or cosine ratios to find the magnitude of each force.
Example 6

34. Right triangle definitions of cosine and sine
Resolve a Force into Rectangular Components
Right triangle definitions of cosine and sine
Solve for x and y.
Use a calculator.
Answer: horizontal component ≈ N; vertical component ≈ N
Example 6

35. SOCCER A player kicks a soccer ball so that it leaves the ground with a velocity of 39 feet per second at an angle of 37° with the ground. Find the magnitude of the horizontal and vertical components of the velocity.
A. horizontal component ≈ ft/s; vertical component ≈ ft/s
B. horizontal component ≈ 37 ft/s; vertical component ≈ 39 ft/s
C. horizontal component ≈ ft/s; vertical component ≈ ft/s
D. horizontal component ≈ ft/s; vertical component ≈ ft/s
Example 6