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Math Help - Vector Direction

  1. #1
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    Vector Direction

    Let the vector

    w = n x (v x n)

    where v is arbitrary and n is a unit vector. In what direction does w point and how is it related to v?
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  2. #2
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    imagine v as the sum of 2 vectors, 1 parallel and 1 perpendicular to n

    then try to imagine the cross product with each one
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  3. #3
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    Quote Originally Posted by maibs89 View Post
    Let the vector

    w = n x (v x n)

    where v is arbitrary and n is a unit vector. In what direction does w point and how is it related to v?
    Note that v x n is perpendicular to v and n. Hence n x (v x n) is the cross product of two orthogonal vectors, and thus has magnitude |n| |v x n| or just |v x n|.

    Next, observe that w is perpendicular to v x n. Since v x n is in itself perpendicular to both v and n, w is on the plane spanned by v and n. Since it is perpendicular to n too, it is a vector of magnitude |v x n| on the plane spanned by v and n, that is perpendicular to n. Which side it's on is
    determined by the "right-hand rule" for cross products.

    Note we assume v and n are not parallel, but if they are, w is just 0.
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  4. #4
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    Quote Originally Posted by maibs89 View Post
    Let the vector

    w = n x (v x n)

    where v is arbitrary and n is a unit vector. In what direction does w point and how is it related to v?
    This is a basic and widely used identity.
    For any vectors a,~b~\&~c~\Rightarrow~a\times (b \times c)=(a\cdot c)b- (a\cdot b)c
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  5. #5
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    Thanks!
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