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.
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$\begingroup$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).
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