Snippets Math / Cross Product

Cross Product

By Marcelo Fernandes Jul 09, 2017

Definition:

Cross Product or Vector Product is a binary operation on two vectors in three-dimensional space, and is often denoted by the symbol "×".

Given two linearly independent vectors a and b, the cross product a × b is a vector that is perpendicular to both a and b and thus normal to the plane containing them.

Geometric Meaning:

The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides, look at the following pic:

Calculation:

The easiest way to calculate a Cross Product is probably by getting the formal determinant of:


Example:

Calculate the area of a triangle represented by the points A(2,3,4), B(1,5,6) and C(4,2,3).

Step 1:
Given the points, get the vectors
AB = B - A = (-1,2,2)
AC = C - A = (2,-1,-1)

Step 2:
The area of a triangle is base*height/2
translating this to vector notation:
Area = 1/2|AB×AC|

Step 3:
Applying on the determinant formula we end up with the determinant: -3k + 3j
the module is equal to = sqrt(18)

and our answer is: 1/2sqrt(18)

Notes


Other Formulas:



References:


1 - Wikipedia