Question

Let x,yER^3 and assume x is not a multiple of y. Show that x (times) y is orthogonal both to x and y

Answer 1

Two vectors are orthogonal if their dot (scalar) product is 0. So you need to check that \((x \times y) \cdot x = 0\) and \((x \times y) \cdot y = 0\).

Answer 2

X x1 y1
Y x2 y2
Z x3 y3


X = x2 y3 - x3 y2
Y = x3 y1 - x1 y3
Z = x1 y2 - x2 y1

dot that with x and y to determine a 0 dot product