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## Study23 Group Title Stuck for a long time :(..........Please help me with vector addition! Illustration below. I need to find the direction and magnitude of the sum vector. The correct answer is 41.62, 42 degrees from x to -y 9 months ago 9 months ago

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1. Study23 Group Title

|dw:1381109606282:dw| (Construct c = a + b)

2. Study23 Group Title

|dw:1381109717758:dw|

3. wolfe8 Group Title

I believe you are supposed to add them like this: |dw:1381110194843:dw|

4. Study23 Group Title

?

5. Study23 Group Title

I was trying to use the component method (x and y components)

6. mathstudent55 Group Title

What are the magnitudes of the two vectors?

7. Study23 Group Title

Oh! I forgot to write that! One second...

8. Study23 Group Title

|dw:1381111020150:dw|

9. mathstudent55 Group Title

|dw:1381111017373:dw|

10. Study23 Group Title

255 or 75?

11. mathstudent55 Group Title

$$R_x = A_x + B_x = 5 \cos20 + 5 \cos255$$ $$R_y = A_y + B_y = 5 \sin20 + 5 \sin255$$ $$R = \sqrt{R_x^2 + R_y^2}$$ $$\theta = \tan^{-1} \dfrac{R_y}{R_x}$$

12. mathstudent55 Group Title

An angle of 75 deg down form the negative y-axis is an angle of 255 deg from the positive x-axis. The advantage of using the 255 deg angle is that the sin and cos will automatically be negative numbers since it's in the third quadrant.

13. Study23 Group Title

Oh!! Okay That makes sense!

14. Study23 Group Title

@mathstudent55 do you mind If I calculate that out to see if I get the correct answer? I've been half an hour, so please don't leave :)

15. mathstudent55 Group Title

No problem.

16. Study23 Group Title

Hmmm I get 4.61 for magnitude?

17. Study23 Group Title

7 degrees?

18. mathstudent55 Group Title

I got 4.62 magnitude, and -42.5 deg.

19. mathstudent55 Group Title

What did you get for Rx and Ry?

20. Study23 Group Title

Rx: $$\ 5cos20+5cos255$$ $$\ =$$ $$\ -0.13...$$ Rx: $$\ 5sin20 + 5sin255$$ $$\ =$$ $$\ 0.416$$

21. Study23 Group Title

$$\ \huge \text{The latter should be } Ry.$$

22. mathstudent55 Group Title

This is what I get for the components of the resultant. $$R_x = 5 \cos 20 + 5 \cos 255 = 4.6985 + (-1.29409) = 3.4044$$ $$R_y = 5 \sin 20 + 5 \sin 255 = 1.7101 + (-4.82963) = -3.11952$$

23. mathstudent55 Group Title

$$R^2 = R_x^2 +R_y^2$$ $$R = \sqrt{(3.4044)^2 + (-3.11952)^2}$$ $$R = 4.6175$$ $$\theta = \tan^{-1}\dfrac{R_y}{R_x}$$ $$\theta = \tan^{-1} \dfrac{-3.11952}{3.4044}$$ $$\theta = 42.5^o$$

24. Study23 Group Title

Hmmm So perhaps the answer key is incorrect?

25. Study23 Group Title

I hope this not too much, but there is one other problem I am stuck on... this time with vector subtraction. Do you mind helping me with that?

26. Study23 Group Title

|dw:1381113129371:dw|

27. mathstudent55 Group Title

a - b = a + (-b)

28. mathstudent55 Group Title

The new vector is -b. Now find x and y components of vector a and vector -b. Add the components together and find the magnitude of the resultant. Then use the inverse tangent to find the angle of the resultant. |dw:1381113881005:dw|

29. mathstudent55 Group Title

|dw:1381114004562:dw|

30. Study23 Group Title

Okay, and I would still add the components except add the OPPOSITE of the b components?

31. Study23 Group Title

would x still be cos and y still be sin in this case? I always forget when those are switched

32. mathstudent55 Group Title

$$A_x = 8 \cos 135$$ $$A_y = 8 \sin 135$$ $$B_x = 4.5 \cos 240$$ $$B_y = 4.5 \sin 240$$ $$R_x = A_x + B_x = 8 \cos 135 + 4.5 \cos 240 = -7.90685$$ $$R_y = A_y + B_y = 8 \sin 135 + 4.5 \sin 240 = 1.75974$$

33. Study23 Group Title

Where do the 135 and 240 come from?

34. Study23 Group Title

It's subtraction, so shouldn't it be - 4.5cos240?

35. mathstudent55 Group Title

Let me explain that again.

36. mathstudent55 Group Title

|dw:1381114977171:dw|

37. mathstudent55 Group Title

These are the original vectors A and B, ok?

38. Study23 Group Title

k

39. mathstudent55 Group Title

|dw:1381115051999:dw|

40. mathstudent55 Group Title

Instead of A + B, they want A - B, right?

41. Study23 Group Title

Yup

42. mathstudent55 Group Title

Mathematically speaking, A + B is the same as A + (-B). For example, 4 - 3 = 4 + (-3)

43. Study23 Group Title

Yes

44. mathstudent55 Group Title

So in order to perform the subtraction of vectors A - B, we can instead add vectors A and -B.

45. Study23 Group Title

Okay

46. mathstudent55 Group Title

The first step now is to find what the vector -B is.

47. Study23 Group Title

k

48. mathstudent55 Group Title

If vector $$B = B_x + B_y$$, then $$-B = -( B_x + B_y) = - B_x + (- B_y)$$

49. mathstudent55 Group Title

If B is the vector in the figure below, |dw:1381115308130:dw|

50. Study23 Group Title

uhuh

51. mathstudent55 Group Title

|dw:1381115353688:dw|

52. mathstudent55 Group Title

Those are the components of B. Then this is -B: |dw:1381115399994:dw|

53. Study23 Group Title

ok

54. mathstudent55 Group Title

Since vector B goes up to the right, vector -B goes down to the left.

55. mathstudent55 Group Title

Now remember that we need to add vectors A and -B to subtract A - B.

56. mathstudent55 Group Title

|dw:1381115515926:dw|

57. mathstudent55 Group Title

There you have vector A and vector -B. Now we need to add them.

58. mathstudent55 Group Title

First, let's translate the angles of the vectors to angles starting at the positive x axis.

59. mathstudent55 Group Title

|dw:1381115630429:dw|

60. Study23 Group Title

okay

61. mathstudent55 Group Title

Now we need to add a vector with magnitude 8 at 135 degrees and a vector of magnitude 4.5 at 240 degrees.

62. Study23 Group Title

k

63. mathstudent55 Group Title

Now we get x and y components of both vectors.

64. Study23 Group Title

ok

65. mathstudent55 Group Title

$$A_x = 8 \cos 135 = -5.65685$$ $$A_y = 8 \sin 135 = 5.65685$$ $$B_x = 4.5 \cos 240 = -2.25$$ $$B_y = 4.5 \sin 240 = -3.89711$$

66. Study23 Group Title

Ahhh! That makes more sense!

67. mathstudent55 Group Title

Now to find the resultant we add the x components to find the x component of the resultant, and we do the same for the y components.

68. mathstudent55 Group Title

$$R_x = A_x + B_x = -5.65685 + (-2.25) = -7.90685$$ $$R_y = A_y + B_y = 5.65685 + (-3.89711) = 1.75974$$

69. mathstudent55 Group Title

Now that we have the x and y components of the resultant, we can use the Pythagorean theorem to find the magnitude of the resultant. $$R = \sqrt{R_x^2 + R_y^2}$$ $$R = 8.1$$ We find the angle by using the inverse tangent: $$\theta = \tan^{-1} \dfrac{R_y}{R_x}$$ $$\theta = \tan^{-1} \dfrac{1.75974}{-7.90685}$$ $$\theta = -12.5^o$$

70. mathstudent55 Group Title

|dw:1381116568714:dw|

71. Study23 Group Title

Thank you so much @mathstudent55! You're help is invaluable! I was stuck on these for soooo long, thank you thank you!

72. mathstudent55 Group Title

You're welcome. BTW, notice how I drew the direction in the last picture. The direction is 12.5 deg up from the negative x-axis, or 167.5 deg measured counterclockwise from the positive x-axis. The answer is 8.1 at 167.5 deg.