How can scalar multiplication of two vectors be intuitively explained in geometrical terms? What does it mean geometrically that the scalar multiplication of two vectors has certain value?
Specifically, I'm asking in relation to the calculation of a distance of certain point from a plane.
Given a following plane: $$A(x-x_0)+B(y-y_0) + C(z-z_0)=0$$ the distance from a point: $$P = (a, b, c)$$ would be $$\frac{|[(a-x_0), (b-y_0), (c-z_0)]\circ[A, B, C]|}{||[A, B,C]||}$$ I don't understand how does it work though. Hence the question.
EDIT
Same thing applies to e.g. calculation of the distance of a line from another line, so probably a general question is:
How does projection work?
as the equation given above is a projection of $$[(a-x_0), (b-y_0), (c-z_0)]$$ through $$[A, B, C]$$