Question

What does the cross product between two vectors represent, and what are some of its properties

Answer 1

Answer:

See explanation below.

Step-by-step explanation:

Definition

The cross product is a binary operation between two vectors defined as following:

Let two vectors $a = (a_1 ,a_2,a_3) , b=(b_1, b_2, b_3)$

The cross product is defined as:

$a x b = (a_2 b_3 -a_3 b_2, a_3 b_1 -a_1 b_3 ,a_1 b_2 -a_2 b_1)$

The last one is the math definition but we have a geometric interpretation as well.

We define the angle between two vectors a and b $\theta$ and we assume that $0\leq \theta \leq \pi$ and we have the following equation:

$|axb| = |a| |b| sin(\theta)$

And then we conclude that the cross product is orthogonal to both of the original vectors.

Some properties

Let a and b vectors

If two vectors a and b are parallel that implies $|axb| =0$

If $axb \neq 0$ then $axb$ is orthogonal to both a and b.

Let u,v,w vectors and c a scalar we have:

$uxv =-v xu$

$ux (v+w) = uxv + uxw$ (Distributive property)

$(cu)xv = ux(cv) =c (uxv)$

$u. (vxw) = (uxv).w$

Other application of the cross product are related to find the area of a parallelogram for two dimensions where:

$A = |axb|$

And when we want to find the volume of a parallelepiped in 3 dimensions:

$V= |a. (bxc)|$