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30 July, 09:19

You are given vectors A = 5.0 ˆi - 6.5 ˆj and B = - 3.5 ˆi + 7.0 ˆj. A third vector C lies in the xy-plane. Vector C is perpendicular to vector A, and the scalar (dot) product of C with B is 15.0. From this information, find the components of vector C.

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  1. 30 July, 09:45
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    Answer: 4.29 and - 3.23

    Explanation: Let vector C be represented as

    C = ai + bj

    First sentence is that vector A is perpendicular to vector C.

    When 2 vectors are perpendicular, their dot product is zero, that is

    A. C = 0

    By doing so, we have that

    A = 5i - 6.5j and C = ai + bj

    A. C = (5i - 6.5j) * (ai + bj) = 0

    5a (i. i) + 5b (i. j) - 6.5a (j. i) - 6.5b (j. j)

    From vector dot product, i. i = j. j = 1 and i. j = j. i = 0

    Hence we have that

    A. C = 5a - 6.5b=0 ... equation 1

    The second sentence is that the dot product of C and B is 15.0

    C. B = (ai + bj) * (-3.5i + 7j) = 15

    C. B = - 3.5a (i. i) + 7a (i. j) - 3.5b (i. j) + 7b (j. j)

    But i. i = j. j = 1 and i. j = j. i = 0

    C. B = - 3.5a + 7b = 15

    -3.5a + 7b = 15 ... equation 2

    5a - 6.5b=0

    -3.5a + 7b = 15

    From the first equation, we make 'a' the subject of formulae and we have that

    5a = 6.5b

    a = 6.5b/5 ... equation 3

    Let us substitute the equation above into equation 2, we have that

    -3.5 (6.5b/5) + 7b = 15

    -22.75b/5 + 7b = 15

    -4.55b = 15

    b = 15 / - 4.55 = - 3.23.

    Substitute b = - 3.23 into equation 3 to get the value for a

    a = 6.5b/5

    a = 6.5 (-3.23) / 5

    a = 21.43/5 = 4.29

    So the x component of (a) of vector C is 4.29 and the y component (b) is - 3.23
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