1. Splash Screen

2. Five-Minute Check (over Lesson 8–6) CCSS Then/Now New Vocabulary
Example 1: Represent Vectors Geometrically
Key Concept: Vector Addition
Example 2: Find the Resultant of Two Vectors
Example 3: Write a Vector in Component Form
Example 4: Find the Magnitude and Direction of a Vector
Key Concept: Vector Operations
Example 5: Operations with Vectors
Example 6: Real-World Example: Vector Applications
Lesson Menu

3. Find s if the measures of ΔRST are mR = 63, mS = 38, and r = 52.
B. 44.6
C. 39.3
D. 35.9
5-Minute Check 1

4. Find mR if the measures of ΔRST are mS = 122, s = 10.8, and r = 5.2.
B. 24.1
C. 29
D. 58
5-Minute Check 2

5. Use the measures of ΔABC to find c to the nearest tenth.
5-Minute Check 3

6. Use the measures of ΔABC to find mB to the nearest degree.
5-Minute Check 4

7. On her delivery route, Gina drives 15 miles west, then makes a 68° turn and drives southeast 14 miles. When she stops, approximately how far from her starting point is she?
A. 21 mi
B. 18 mi
C. 16 mi
D mi
5-Minute Check 5

8. Mathematical Practices
Content Standards
G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
4 Model with mathematics.
CCSS

9. Perform vector operations geometrically.
You used trigonometry to find side lengths and angle measures of right triangles.
Perform vector operations geometrically.
Perform vector operations on the coordinate plane.
Then/Now

10. vector magnitude direction resultant parallelogram method
triangle method
standard position
component form
Vocabulary

11. = 80 meters at 24° west of north
Represent Vectors Geometrically
A. Use a ruler and a protractor to draw each vector. Include a scale on each diagram.
= 80 meters at 24° west of north
Using a scale of 1 cm : 50 m, draw and label an 80 ÷ 50 or 1.6-centimeter arrow 24º west of the north-south line on the north side.
Answer:
Example 1

12. = 16 yards per second at 165° to the horizontal
Represent Vectors Geometrically
B. Use a ruler and a protractor to draw each vector. Include a scale on each diagram.
= 16 yards per second at 165° to the horizontal
Using a scale of 1 cm : 8 yd/s, draw and label a 16 ÷ 8 or 2-centimeter arrow at a 165º angle to the horizontal.
Answer:
Example 1

13. Using a ruler and a protractor, draw a vector to represent feet per second 25 east of north. Include a scale on your diagram. A. B. C. D.
Example 1

14. Concept

15. Copy the vectors. Then find
Find the Resultant of Two Vectors
Copy the vectors. Then find a b
Subtracting a vector is equivalent to adding its opposite.
Example 2

16. Method 1 Use the parallelogram method.
Find the Resultant of Two Vectors
Method 1 Use the parallelogram method.
Step , and translate it so that its tail touches the tail of . –b a
Example 2

17. Step 2 Complete the parallelogram. Then draw the diagonal.
Find the Resultant of Two Vectors
Step 2 Complete the parallelogram. Then draw the diagonal.
a – b –b a
Example 2

18. Method 2 Use the triangle method.
Find the Resultant of Two Vectors
Method 2 Use the triangle method.
Step , and translate it so that its tail touches the tail of . –b a
Example 2

19. Step 2 Draw the resultant vector from the tail of to the tip of – .
Find the Resultant of Two Vectors
Step 2 Draw the resultant vector from the tail of to the tip of – . a –b
a – b
Answer:
a – b
Example 2

20. Copy the vectors. Then findb a
A. B.
C. D.
a – b
Example 2

21. Write the component form of .
Write a Vector in Component Form
Write the component form of
Example 3

22. Find the change of x-values and the corresponding change in y-values.
Write a Vector in Component Form
Find the change of x-values and the corresponding change in y-values.
Component form of vector
Simplify.
Example 3

23. Write the component form of .A. B. C. D.
Example 3

24. Find the magnitude and direction of
Find the Magnitude and Direction of a Vector
Find the magnitude and direction of
Step 1 Use the Distance Formula to find the vector’s magnitude.
Distance Formula
(x1, y1) = (0, 0) and (x2, y2) = (7, –5)
Simplify.
Use a calculator.
Example 4

25. Step 2 Use trigonometry to find the vector’s direction.
Find the Magnitude and Direction of a Vector
Step 2 Use trigonometry to find the vector’s direction.
Graph , its horizontal component, and its vertical component. Then use the inverse tangent function to find θ.
Example 4

26. Definition of inverse tangent
Find the Magnitude and Direction of a Vector
Definition of inverse tangent
Use a calculator.
The direction of is the measure of the angle that it makes with the positive x-axis, which is about 360 – 35.5 or So, the magnitude of is about 8.6 units and the direction is at an angle of about 324.5º to the horizontal.
Answer:
Example 4

27. Find the magnitude and direction of
B. 5.7; 45°
C. 5.7; 225°
D. 8; 135°
Example 4

28. Concept

29. Operations with Vectors
Find each of the following for and Check your answers graphically. A.
Solve Algebraically
Check Graphically
Example 5

30. Operations with Vectors
Find each of the following for and Check your answers graphically. B.
Solve Algebraically
Check Graphically
Example 5

31. Operations with Vectors
Find each of the following for and Check your answers graphically. C.
Solve Algebraically
Check Graphically
Example 5

32. A. B. C. D.
Example 5

33. Draw a diagram. Let represent the resultant vector.
Vector Applications
CANOEING Suppose a person is canoeing due east across a river at 4 miles per hour. If the river is flowing south at 3 miles per hour, what is the resultant speed and direction of the canoe?
Draw a diagram. Let represent the resultant vector.
Example 6

34. Vector Applications
The component form of the vector representing the velocity of the canoe is 4, 0, and the component form of the vector representing the velocity of the river is 0, –3. The resultant vector is 4, 0 + 0, –3 or 4, –3, which represents the resultant velocity of the canoe. Its magnitude represents the resultant speed.
Example 6

35. Use the Distance Formula to find the resultant speed.
Vector Applications
Use the Distance Formula to find the resultant speed.
Distance Formula
(x1, y1) = (0, 0) and (x2, y2) = (4, –3)
The resultant speed of the canoe is 5 miles per hour.
Example 6

36. Use trigonometry to find the resultant direction.
Vector Applications
Use trigonometry to find the resultant direction.
Definition of inverse tangent
Use a calculator.
The resultant direction of the canoe is about 36.9° south of due east.
Answer: Therefore, the resultant speed of the canoe is 5 mile per hour at an angle of about 90° – 36.9° or 53.1° east of south.
Example 6

37. KAYAKING Suppose a person is kayaking due east across a lake at 7 miles per hour. If the lake is flowing south at 4 miles an hour, what is the resultant direction and speed of the canoe?
A. Direction is about 60.3° south of due east with a velocity of about 8.1 miles per hour.
B. Direction is about 60.3° south of due east with a velocity of about 11 miles per hour.
C. Direction is about 29.7° south of due east with a velocity of about 8.1 miles per hour.
D. Direction is about 29.7° south of due east with a velocity of about 11 miles per hour.
Example 6

38. End of the Lesson