cross product-睇唔明solution help

Firstly, normal vector is given by the vector that is perpendicular to the plane.

Given A, B, C 3 points on the plane, AB and AC (if not parallel) is two vector that determines the direction of the plane.

i.e., Let AB = x, AC = y, and the coordinate of A is a (<- all vectors)
Any point on the plane can be expressed by r = a + bx + cy where b and c are real numbers.

If we take AB x AC to get the normal vector n, it must be perpendicular to the plane. Why?

Put the normal vector on a point of the plane. Let the point be a + bx + cy.

We compare this vector with the vector from A to that point. i.e., bx + cy.

Take the dot product between these two vector:

Denote <a,b> be the dot product of a and b (well, inner product)

<bx+cy, n>
= <bx,n> + <cy,n>
= b<x,n> + c<y,n>
= b*0 + c*0 (because n is perpendicular to x and y, by definition of cross prod)
= 0

Therefore bx + cy and the normal vector is perpendicular.

The concept 'cross product' only exist in 3D geometry because in higher dimension this concept is vague (e.g. you have infinite possibilities of unit vector perpendicular to two given vector, etc) so I would say yes.

btw UST math1013? lol

2012-11-10 20:27:55 補充：
I don't think I used cross product between points.

Talking about AB x AC I've already denote it as vectors. If it hasn't been clear enough, let's say like that:

Let AB(from A to B) be vector u and AC be vector v. We are considering u x v (u cross v) there.

The provided solution is a mistake. There is no cross product between points, cross product must be for vectors.
Somehow the answer is numerically correct as by coincident the plane passes through the origin, the coordinate for A numerically = vector OA, etc...

2012-11-10 23:13:11 補充：
I am not talking about yours, I am talking the solution in the original question:
http://i987.photobucket.com/albums/ae355/delaynopemore/123.jpg