How to Take Dot Product of a Vector and a Bivector

Question

I'm new to geometric Algebra and im trying to take geometric product of a bivector and a vector I can understand wedge product but I can't get the meaning of a dot product between bivector and a vector.

Answer 1

This formula is related to the cross product bac-cab identity:

$$a\cdot(b\wedge c)=(a\cdot b)c-(a\cdot c)b$$

(To prove this, just verify that it's true for the basis vectors $e_i$, and it extends by linearity to all vectors.)

This shows that if $a$ is perpendicular to the plane of $b$ and $c$, then the dot product is $0$. It also shows that the result is in the plane, being a combination of $b$ and $c$.

Take the dot product with $a$ again

$$a\cdot\big(a\cdot(b\wedge c)\big)=(a\cdot b)(a\cdot c)-(a\cdot c)(a\cdot b)=0$$

to see that the result is always perpendicular to $a$.

If $a$ is parallel to the plane, then the bivector acts as an imaginary number; the dot product rotates $a$ by $90^\circ$ and scales by the magnitude of the bivector.

So, for general $a$, the dot product acts as a projection-rotation-scaling (in any order).