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Linear algebra: How can ||u x v||^2 = ||u||^2 * ||v||^2 - (u.v)^2 ?
Linear algebra: How can ||u x v||^2 = ||u||^2 * ||v||^2 - (u.v)^2 ?
I don't understand how these statements are equivalent. Any hints are appreciated. Thanks.
1 Answer
- 1 decade agoFavorite Answer
Think about the definition of dot product and cross product.
The magnitude of the cross product between two vectors u and v is defined as |u| |v| sin (theta), where theta is the angle between the two vectors.
So |u x v|^2 = [|u| |v| sin(theta)]^2 = |u|^2 |v|^2 sin^2(theta).
sin^2(theta) = 1 - cos^2(theta), so
|u|^2 |v|^2 sin^2(theta) = |u|^2 |v|^2 (1-cos^2(theta))
= |u|^2 |v|^2 - (|u| |v| cos(theta))^2
and remember that the definition of dot product is |u| |v| cos(theta), you have:
|u|^2 |v|^2 - (u dot v)^2.
