Cross product

Notes:

Cross Product
Suppose given two vectors $A$ and $B$, you want to find a third vector that is perpendicular to them. To find the third vector, we need to do the cross product.
Let $A= \lt a_1,a_2,a_3 \gt$ and $B= \lt b_1,b_2,b_3 \gt$. Then the cross product of these two vectors will be:

$A \times B = \lt a_2 b_3 - b_2 a_3 , b_1a_3 -a_1b_3 , a_1 b_2 - a_2 b_1 \gt$

The formula is ugly to remember, so we have a technique that we will show you in the introduction videos!

Cross Product is NOT commutative
Keep in mind that

$A \times B \ne B \times A$

However, the relationship between these two cross products is that they are pointing in different directions. Hence,

$A \times B = -(B \times A)$

Properties of Cross Products
Let $u,v$ and $w$ be vectors and $c$ is a scalar. Then we have the following cross product properties:

1. $u \times (v+w) = u \times v + u \times w$
2. $(u + v) \times w = u \times w + v \times w$
3. $(cu) \times v = c(u \times v) = u \times (cv)$
4. $u \cdot (v \times w ) = (u \times v) \cdot w$
5. $u \times (v \times w ) = (u \cdot w) v - (u \cdot v) w$