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

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

by RoRi

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

2 comments

  2. 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

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):