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31 March, 00:52

Find the cross product (7,9,6) x (-4,1,5). Is the resulting vector perpendicular to the given vectors

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  1. 31 March, 01:33
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    Expand each vector as linear combinations of the standard basis vectors:

    (7, 9, 6) = 7 (1, 0, 0) + 9 (0, 1, 0) + 6 (0, 0, 1)

    (-4, 1, 5) = - 4 (1, 0, 0) + (0, 1, 0) + 5 (0, 0, 1)

    For brevity, write

    i = (1, 0, 0)

    j = (0, 1, 0)

    k = (0, 0, 1)

    Then by definition of the cross product,

    i x i = j x j = k x k = (0, 0, 0)

    i x j = k

    j x k = i

    k x i = j

    and for any two vectors a and b, we have a x b = - b x a.

    Now compute the product:

    (7i + 9j + 6k) x (-4i + j + 5k)

    = - 28 (i x i) - 36 (j x i) - 24 (k x i)

    ... + 7 (i x j) + 9 (j x j) + 6 (k x j)

    ... + 35 (i x k) + 45 (j x k) + 30 (k x k)

    = - 36 (-k) - 24 j + 7 k + 6 (-i) + 35 (-j) + 45 i

    = 39 i - 59 j + 43 k

    which is the same as the vector

    (39, - 59, 43)

    And yes, this vector is perpendicular to both given vectors.
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