1. Vectors in the Coordinate Plane LESSON 8–2

2. Lesson Menu Five-Minute Check (over Lesson 8-1) TEKS Then/Now New Vocabulary Key Concept: Component Form of a Vector Example 1:Express a Vector in Component Form Key Concept:Magnitude of a Vector in the Coordinate Plane Example 2:Find the Magnitude of a Vector Key Concept:Vector Operations Example 3:Operations with Vectors Example 4:Find a Unit Vector with the Same Direction as a Given Vector Example 5:Write a Unit Vector as a Linear Combination of Unit Vectors Example 6:Find Component Form Example 7:Direction Angles of Vectors Example 8:Real-World Example: Applied Vector Operations

3. Over Lesson 8-1 5–Minute Check 1 Determine the magnitude and direction of the resultant of the vector sum described as 25 miles east and then 47 miles south. A.53.2 miles, N28°E B.53.2 miles, S62°E C.53.2 miles, S28°E D.72 miles, S28°E

4. Over Lesson 8-1 5–Minute Check 2 An airplane heads due east at 400 miles per hour. If a 40-mile-per-hour wind blows from a bearing of N 25°E, what are the ground speed and direction of the plane? A.418.5 mph, N85°E B.418.5 mph, N5°E C.384.8 mph, S5.4°E D.384.8 mph, S84.6°E

5. Over Lesson 8-1 5–Minute Check 3 Which of the following represents a vector quantity? A.a car driving at 55 miles per hour B.a cart pulled up a 30° incline with a force of 40 newtons C.the temperature of a cup of coffee D.wind blowing at 30 knots

6. Targeted TEKS P.4(J) Represent the addition of vectors and the multiplication of a vector by a scalar geometrically and symbolically. P.4(K) Apply vector addition and multiplication of a vector by a scalar in mathematical and real-world problems. Mathematical Processes P.1(B), P.1(F)

7. Then/Now You performed vector operations using scale drawings. (Lesson 8-1) Represent and operate with vectors in the coordinate plane. Write a vector as a linear combination of unit vectors.

8. Vocabulary component form unit vector linear combination

9. Key Concept 1

10. Example 1 Express a Vector in Component Form Find the component form of with initial point A(1, –3) and terminal point B(1, 3). =  x 2 – x 1, y 2 – y 1  Component form =  1 – 1, 3 – (–3)  (x 1, y 1 ) = (1, –3) and ( x 2, y 2 ) = (1, 3) =  0, 6  Subtract. Answer:  0, 6 

11. Example 1 A.  9, 6  B.  6, 9  C.  1, 0  D.  0, 1  Find the component form of given initial point A(–4, –3) and terminal point B(5, 3).

12. Key Concept 2

13. Example 2 Find the Magnitude of a Vector Find the magnitude of with initial point A(1, –3) and terminal point B(1, 3). Simplify. (x 1, y 1 ) = (1, –3) and ( x 2, y 2 ) = (1, 3) Magnitude formula

14. Example 2 Find the Magnitude of a Vector Answer: 6 CHECK From Example 1, you know that =  0, 6 .

15. Example 2 A.1 B.7 C.7.3 D.23 Find the magnitude of given initial point A(4, –2) and terminal point B(–3, –2).

16. Key Concept 3

17. Example 3 Operations with Vectors A. Find 2w + y for w =  2, –5 , y =  2, 0 , and z =  –1, –4 . 2w + y = 2  2, –5  +  2, 0  Substitute. =  4,  10  +  2, 0  Scalar multiplication =  4 + 2, –10 + 0  or  6, –10  Vector addition Answer:  6, –10 

18. Example 3 Operations with Vectors B. Find 3y – 2z for w =  2, –5 , y =  2, 0 , and z =  –1, –4 . 3y – 2z= 3y + (–2z)Rewrite subtraction as addition. = 3  2, 0  + (–2)  –1, –4  Substitute. =  6, 0  +  2, 8  Scalar multiplication =  6 + 2, 0 + 8  or  8, 8  Vector addition Answer:  8, 8 

19. Example 3 Find 3v + 2w for v =  4, –1  and w =  –3, 5 . A.  18, 13  B.  6, 7  C.  6, 13  D.  –1, 13 

20. Example 4 Find a Unit Vector with the Same Direction as a Given Vector Find a unit vector u with the same direction as v =  4, –2 . Unit vector with the same direction as v. Substitute. ; Simplify. or

21. Example 4 Find a Unit Vector with the Same Direction as a Given Vector Rationalize the denominator. Scalar multiplication Rationalize denominators. Therefore, u =. Answer:u =

22. Example 4 Find a Unit Vector with the Same Direction as a Given Vector Check Since u is a scalar multiple of v, it has the same direction as v. Verify that the magnitude of u is 1. Magnitude Formula Simplify. Simplify.

23. Example 4 Find a unit vector u with the same direction as w =  5, –3 . A. B. C. D.

24. Example 5 Write a Unit Vector as a Linear Combination of Unit Vectors First, find the component form of. Let be the vector with initial point D(–3, –3) and terminal point E(2, 6). Write as a linear combination of the vectors i and j. =  x 2 – x 1, y 2 – y 1  Component form =  2 – (–3), 6 – (–3)  (x 1, y 1 ) = (–3, –3) and ( x 2, y 2 ) = (2, 6) =  5, 9  Subtract.

25. Example 5 Answer:5i + 9j Write a Unit Vector as a Linear Combination of Unit Vectors Then, rewrite the vector as a linear combination of the standard unit vectors. =  5, 9  Component form = 5i + 9j  a, b  = ai + bj

26. Example 5 A.2i + 3j B.3i + 8j C.5i + 2j D.3i + 2j Let be the vector with initial point D(–4, 3) and terminal point E(–1, 5). Write as a linear combination of the vectors i and j.

27. Example 6 Find the component form of the vector v with magnitude 7 and direction angle 60°. Find Component Form Component form of v in terms of |v| and θ |v| = 7 and θ = 60° Simplify. Answer:

28. Example 6 Find Component Form Check Graph v = ≈  3.5, 6.1 . The measure of the angle v makes with the positive x-axis is about 60° as shown, and |v| =.

29. Example 6 Find the component form of the vector v with magnitude 12 and direction angle 300°. A. B. C. D.

30. Example 7 Direction Angles of Vectors A. Find the direction angle of p =  2, 9  to the nearest tenth of a degree. Direction angle equation a = 2 and b = 9 Solve for . Use a calculator.

31. Example 7 Direction Angles of Vectors Answer: 77.5° So the direction angle of vector p is about 77.5°, as shown below.

32. Example 7 Direction Angles of Vectors B. Find the direction angle of r = –7i + 2j to the nearest tenth of a degree. Use a calculator. a = –7 and b = 2 Direction angle equation Solve for .

33. Example 7 Direction Angles of Vectors Answer: 164.1° Since r lies in Quadrant II as shown below,  = 180 – 15.9° or 164.1°.

34. Example 7 Find the direction angle of p =  –1, 4  to the nearest tenth of a degree. A.14.5° B.76.3° C.104.5° D.166.7°

35. Example 8 SOCCER A soccer player running forward at 7 meters per second kicks a soccer ball with a velocity of 30 meters per second at an angle of 10° with the horizontal. What is the resultant speed and direction of the kick? Since the soccer player moves straight forward, the component form of his velocity v 1 is  7, 0 . Use the magnitude and direction of the soccer ball’s velocity v 2 to write this vector in component form. Applied Vector Operations

36. Example 8 v 2 =  | v 2 | cos θ, | v 2 | sin θ  Component form of v 2 =  30 cos 10°, 30 sin 10°  |v 2 | = 30 and θ = 10° ≈  29.5, 5.2  Simplify. Add the algebraic vectors representing v 1 and v 2 to find the resultant velocity, r. r= v 1 + v 2 Resultant vector =  7, 0  +  29.5, 5.2  Substitution =  36.5, 5.2  Vector Addition Applied Vector Operations

37. Example 8 Applied Vector Operations The magnitude of the resultant is |r| = or about 36.9. Next find the resultant direction θ.

38. Example 8 Answer:36.9 m/s; 8.1° Applied Vector Operations Therefore, the resultant velocity of the kick is about 36.9 meters per second at an angle of about 8.1° with the horizontal.  a, b  =  36.5, 5.2 

39. Example 8 SOCCER A soccer player running forward at 6 meters per second kicks a soccer ball with a velocity of 25 meters per second at an angle of 15° with the horizontal. What is the resultant speed and direction of the kick? A.25.0 m/s; 15.1° B.25.0 m/s; 8.1° C.30.8 m/s; 15.1° D.30.8 m/s; 12.1°

40. Vectors in the Coordinate Plane LESSON 8–2