1. Splash Screen

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

3. = 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

4. = 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

5. 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

6. Concept

7. 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

8. 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

9. 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

10. 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

11. 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

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

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

14. 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

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

16. 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

17. 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

18. 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

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

20. Concept

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

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

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

24. A. B. C. D.
Example 5

25. 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

26. 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

27. 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

28. 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

29. 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

30. End of the Lesson