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Study23

  • 2 years ago

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

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  1. Study23
    • 2 years ago
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    |dw:1381109606282:dw| (Construct c = a + b)

  2. Study23
    • 2 years ago
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    |dw:1381109717758:dw|

  3. wolfe8
    • 2 years ago
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    I believe you are supposed to add them like this: |dw:1381110194843:dw|

  4. Study23
    • 2 years ago
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    ?

  5. Study23
    • 2 years ago
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    I was trying to use the component method (x and y components)

  6. mathstudent55
    • 2 years ago
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    What are the magnitudes of the two vectors?

  7. Study23
    • 2 years ago
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    Oh! I forgot to write that! One second...

  8. Study23
    • 2 years ago
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    |dw:1381111020150:dw|

  9. mathstudent55
    • 2 years ago
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    |dw:1381111017373:dw|

  10. Study23
    • 2 years ago
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    255 or 75?

  11. mathstudent55
    • 2 years ago
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    \(R_x = A_x + B_x = 5 \cos20 + 5 \cos255 \) \(R_y = A_y + B_y = 5 \sin20 + 5 \sin255 \) \(R = \sqrt{R_x^2 + R_y^2} \) \(\theta = \tan^{-1} \dfrac{R_y}{R_x} \)

  12. mathstudent55
    • 2 years ago
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    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
    • 2 years ago
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    Oh!! Okay That makes sense!

  14. Study23
    • 2 years ago
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    @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
    • 2 years ago
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    No problem.

  16. Study23
    • 2 years ago
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    Hmmm I get 4.61 for magnitude?

  17. Study23
    • 2 years ago
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    7 degrees?

  18. mathstudent55
    • 2 years ago
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    I got 4.62 magnitude, and -42.5 deg.

  19. mathstudent55
    • 2 years ago
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    What did you get for Rx and Ry?

  20. Study23
    • 2 years ago
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    Rx: \(\ 5cos20+5cos255 \) \(\ = \) \(\ -0.13... \) Rx: \(\ 5sin20 + 5sin255 \) \(\ = \) \(\ 0.416 \)

  21. Study23
    • 2 years ago
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    \(\ \huge \text{The latter should be } Ry. \)

  22. mathstudent55
    • 2 years ago
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    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
    • 2 years ago
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    \( R^2 = R_x^2 +R_y^2 \) \( R = \sqrt{(3.4044)^2 + (-3.11952)^2} \) \( R = 4.6175 \) \( \theta = \tan^{-1}\dfrac{R_y}{R_x} \) \( \theta = \tan^{-1} \dfrac{-3.11952}{3.4044} \) \(\theta = 42.5^o\)

  24. Study23
    • 2 years ago
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    Hmmm So perhaps the answer key is incorrect?

  25. Study23
    • 2 years ago
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    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
    • 2 years ago
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    |dw:1381113129371:dw|

  27. mathstudent55
    • 2 years ago
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    a - b = a + (-b)

  28. mathstudent55
    • 2 years ago
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    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
    • 2 years ago
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    |dw:1381114004562:dw|

  30. Study23
    • 2 years ago
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    Okay, and I would still add the components except add the OPPOSITE of the b components?

  31. Study23
    • 2 years ago
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    would x still be cos and y still be sin in this case? I always forget when those are switched

  32. mathstudent55
    • 2 years ago
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    \(A_x = 8 \cos 135\) \(A_y = 8 \sin 135 \) \(B_x = 4.5 \cos 240 \) \( B_y = 4.5 \sin 240 \) \(R_x = A_x + B_x = 8 \cos 135 + 4.5 \cos 240 = -7.90685\) \(R_y = A_y + B_y = 8 \sin 135 + 4.5 \sin 240 = 1.75974\)

  33. Study23
    • 2 years ago
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    Where do the 135 and 240 come from?

  34. Study23
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    It's subtraction, so shouldn't it be - 4.5cos240?

  35. mathstudent55
    • 2 years ago
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    Let me explain that again.

  36. mathstudent55
    • 2 years ago
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    |dw:1381114977171:dw|

  37. mathstudent55
    • 2 years ago
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    These are the original vectors A and B, ok?

  38. Study23
    • 2 years ago
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    k

  39. mathstudent55
    • 2 years ago
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    |dw:1381115051999:dw|

  40. mathstudent55
    • 2 years ago
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    Instead of A + B, they want A - B, right?

  41. Study23
    • 2 years ago
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    Yup

  42. mathstudent55
    • 2 years ago
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    Mathematically speaking, A + B is the same as A + (-B). For example, 4 - 3 = 4 + (-3)

  43. Study23
    • 2 years ago
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    Yes

  44. mathstudent55
    • 2 years ago
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    So in order to perform the subtraction of vectors A - B, we can instead add vectors A and -B.

  45. Study23
    • 2 years ago
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    Okay

  46. mathstudent55
    • 2 years ago
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    The first step now is to find what the vector -B is.

  47. Study23
    • 2 years ago
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    k

  48. mathstudent55
    • 2 years ago
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    If vector \(B = B_x + B_y\), then \(-B = -( B_x + B_y) = - B_x + (- B_y)\)

  49. mathstudent55
    • 2 years ago
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    If B is the vector in the figure below, |dw:1381115308130:dw|

  50. Study23
    • 2 years ago
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    uhuh

  51. mathstudent55
    • 2 years ago
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    |dw:1381115353688:dw|

  52. mathstudent55
    • 2 years ago
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    Those are the components of B. Then this is -B: |dw:1381115399994:dw|

  53. Study23
    • 2 years ago
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    ok

  54. mathstudent55
    • 2 years ago
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    Since vector B goes up to the right, vector -B goes down to the left.

  55. mathstudent55
    • 2 years ago
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    Now remember that we need to add vectors A and -B to subtract A - B.

  56. mathstudent55
    • 2 years ago
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    |dw:1381115515926:dw|

  57. mathstudent55
    • 2 years ago
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    There you have vector A and vector -B. Now we need to add them.

  58. mathstudent55
    • 2 years ago
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    First, let's translate the angles of the vectors to angles starting at the positive x axis.

  59. mathstudent55
    • 2 years ago
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    |dw:1381115630429:dw|

  60. Study23
    • 2 years ago
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    okay

  61. mathstudent55
    • 2 years ago
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    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
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    k

  63. mathstudent55
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    Now we get x and y components of both vectors.

  64. Study23
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    ok

  65. mathstudent55
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    \(A_x = 8 \cos 135 = -5.65685\) \(A_y = 8 \sin 135 = 5.65685\) \(B_x = 4.5 \cos 240 = -2.25\) \(B_y = 4.5 \sin 240 = -3.89711 \)

  66. Study23
    • 2 years ago
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    Ahhh! That makes more sense!

  67. mathstudent55
    • 2 years ago
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    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
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    \( R_x = A_x + B_x = -5.65685 + (-2.25) = -7.90685 \) \(R_y = A_y + B_y = 5.65685 + (-3.89711) = 1.75974 \)

  69. mathstudent55
    • 2 years ago
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    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
    • 2 years ago
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    |dw:1381116568714:dw|

  71. Study23
    • 2 years ago
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    Thank you so much @mathstudent55! You're help is invaluable! I was stuck on these for soooo long, thank you thank you!

  72. mathstudent55
    • 2 years ago
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    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.

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