Why is $a\times b = 0$ when $b = 2*a$?

Question

In vector calculus, why is $a\times b = 0$ when you know that $b=2*a$? So how how do you know that a crossproduct of a vector and two times that vector is always zero?

Answer 1

The cross product is antisymmetric and bilinear. Antisymmetric means that $$v\times v=0$$ for all vectors $v$. Bilinear implies that $$au\times v=u\times av=a(u\times v)$$ Thus for any scalar $a$ and vector $v$ we have $$v\times av=a(v\times v)=a\cdot 0=0$$

Answer 2

One rational is that you can think of the (norm of the) cross product as giving the area of the paralellagram formed by the two vectors. If you have two vectors that are parallel (or antiparallel) then putting them up against each other gives you no area, as there isn't a second dimension involved.

Answer 3

Since $\lvert a\times b\lvert=\lvert a\lvert \lvert b\lvert\sin\theta$ where $\theta$ is the angle between a and b,

$\lvert a\times b\lvert=0$ and therefore $a\times b=\vec{0}$ whenever a and b are parallel.