Why is the cross product perpendicular?

JizzaDaMan

Why is the cross product of two vectors perpendicular to the plane the two vectors lie on?

I am aware that you can prove this by showing that:

[itex](\vec{a}\times\vec{b})\cdot\vec{a} = (\vec{a}\times\vec{b})\cdot\vec{b} = 0[/itex]

Surely it was not defined as this and worked backwards though. I see little advantage in making this definition, and simply guessing it seems a bit random, so what brings it about?

micromass

What is your definition of the cross product?

Ferramentarius

By the matrix definition of the cross product we have
[itex] \vec{a}\times \vec{b} \cdot \vec{c}
= \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_i & a_j & a_k \\ b_i & b_j & b_k \end{vmatrix} \cdot \vec{c}
= (\vec{i} \begin{vmatrix} a_j & a_k \\ b_j & b_k \end{vmatrix} -\vec{j} \begin{vmatrix} a_i & a_k \\ b_i & b_k \end{vmatrix} + \vec{k} \begin{vmatrix} a_i & a_j \\ b_i & b_j \end{vmatrix} ) \cdot \vec{c} \\
= (c_i \begin{vmatrix} a_j & a_k \\ b_j & b_k \end{vmatrix} -c_j \begin{vmatrix} a_i & a_k \\ b_i & b_k \end{vmatrix} + c_k \begin{vmatrix} a_i & a_j \\ b_i & b_j \end{vmatrix} )
= \begin{vmatrix} c_i & c_j & c_k \\ a_i & a_j & a_k \\ b_i & b_j & b_k \end{vmatrix} [/itex].

When [itex] \vec{c} = \vec{a} [/itex] or [itex] \vec{c} = \vec{b} [/itex] the determinant has two equal rows and becomes zero. This means the dot product is zero and the vectors are perpendicular.

lurflurf

The cross product is the (up to multiplication by a constant) only product possible that takes two vectors to a third. It is also extremely useful to produce a vector perpendicular to two given vectors. All the time you have two vectors and need one perpendicular to them. Bam! Cross product done.