Scalar Triple Product and Definition

Question

I noticed something when I tried to solve the problem today. The scalar three-dimensional product is the same as the determinant or three by three matrices with three vectors in the form of rows:

A = [ a , b , c ]

det (A) = ( a X b ) * c

I met this in Real Timer Rendering, and I can’t understand why this is so, or even if it is useful. It seems that this is due to the short conclusion method of calculating the cross product using a deterministic place where you write unit vectors along the top of the matrix, but I always thought that it was more mnemonic rather than sound mathematics proper.

Is there a real relationship here, or is it just some kind of happy coincidence?

Answer 1

Prior to the sign, the determinant of the n-on-n matrix is ​​the volume of the parallelepiped spanned by its n-dimensional rows (or columns) of vectors (or the volume of a unit cube linearly transformed by this matrix). The product (axb) .c in three dimensions is exactly the same; axb gives a vector perpendicular to a and b, and a length equal to the area of ​​the parallelogram spanned by a and b; (axb) .c gives the height c above this parallelogram, once its area. So no, this is not a coincidence.

Answer 2

; . , - a X b , j k, a1 a2 a3, - b1 b2 b3.

|i  j   k|
|a1 a2 a3|
|b1 b2 b3|

dotproduct c, , c .

|i  j   k|                      |c1 c2 c3|     |c1 c2 c3|    |a1 a2 a3|
|a1 a2 a3| .  (c1,c2,c3)  =     |a1 a2 a3|  = -|a1 a2 a3|  = |b1 b2 b3|
|b1 b2 b3|                      |b1 b2 b3|     |b1 b2 b3|    |c1 c2 c3|

: wikipedia , , Q.E.D.