# How to Take Dot Product of a Vector and a Bivector

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.

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).