- #1
autodidude
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How would you show that the dot product between the normal unit vector of a plane and a position vector to any point on the plane is always the same without using this formula
[tex]n.(r-r_0) = 0[/tex]
∴ [tex]n.r=n.r_0[/tex]
where [tex]n[/tex] is the normal vector, [tex]r[/tex] and [tex]r_o[/tex] are two position vectors to two points on the plane.
I'm looking for an alternative geometric argument/proof that applies to all cases.
I notice that if you have a point P on a plane that is directly above the origin which is parallel to the xy-plane, then the dot product is simply the magnitude of the vector OP. Then as you move further out from the origin to some point [tex]P_n[/tex] on the plane, the position vector gets larger and the projection of the unit normal vector on the vector [tex]OP_n[/tex] gets smaller. One gets larger, the other gets smaller and somehow their product is always the same.
So again, I'm after a way to prove this for all cases geometrically.
Thanks
[tex]n.(r-r_0) = 0[/tex]
∴ [tex]n.r=n.r_0[/tex]
where [tex]n[/tex] is the normal vector, [tex]r[/tex] and [tex]r_o[/tex] are two position vectors to two points on the plane.
I'm looking for an alternative geometric argument/proof that applies to all cases.
I notice that if you have a point P on a plane that is directly above the origin which is parallel to the xy-plane, then the dot product is simply the magnitude of the vector OP. Then as you move further out from the origin to some point [tex]P_n[/tex] on the plane, the position vector gets larger and the projection of the unit normal vector on the vector [tex]OP_n[/tex] gets smaller. One gets larger, the other gets smaller and somehow their product is always the same.
So again, I'm after a way to prove this for all cases geometrically.
Thanks