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21 July, 09:52

Show that (*) * = * (*) if and only if the vectors and are collinear.

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  1. 21 July, 10:54
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    Step-by-step explanation:

    NOTE: Bolded letters are vectors

    Given that,

    (axb) xc = ax (bxc)

    From vector triple product rule,

    ax (bxc) = b (a. c) - c (a. b) ax (bxc) = b (a. c) - a (b. c)

    If these both are equal, then:

    ⇒b (a. c) - c (a. b) = b (a. c) - a (b. c)

    ⇒ (-a. b) c = (-b. c) a

    ⇒ (|a||b|sin∅) c = (|b||c|sinФ) a (cancel |b| on both sides)

    A, C are the unit vectors of a and c respectively.

    Two vectors can be equal on when their magnitude and directions are same.

    That is when,

    ⇒ (|a||c|sin∅) A = (|a||c|sinФ) C

    ⇒sin∅A = sinФC

    This can only be possible when the vectors are collinear because the angle made by the vectors with the vector B should be the same or supplementary and the vectors A and C must be in the same direction.
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