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Prove that the norm of the cross product is the product of the norms if and only if A and B are orthogonal

Given vectors A,B prove that

    \[ \lVert A \times B \rVert =\lVert A \rVert \lVert B \rVert \]

if and only if A and B are orthogonal.


Proof.

From Theorem 13.12(f) (page 483 of Apostol) we know

    \[ \lVert A \times B \rVert^2 = \lVert A \rVert^2 + \lVert B \rVert^2 - (A \cdot B)^2. \]

But (A \cdot B)^2 = 0 if and only if A and B are orthogonal (from the definition of orthogonality). Thus, \lVert A \times B \rVert = \lVert A \rVert \lVert B \rVert if and only if A and B are orthogonal. \qquad \blacksquare

One comment

  1. Anonymous says:

    There is a typo in the formula as stated. Instead of adding the squared norms of A and B, you need to multiply them instead

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