# Cross product

### Cross product

#### Lessons

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. $u \times (v \times w ) = (u \cdot w) v - (u \cdot v) w$
• Introduction
Cross Product Overview
a)
Cross Product
• A vector perpendicular to the other two
• Formula for the cross product
• Technique to use cross product

b)
Cross Product is Not Commutative
• Order in which you do cross product matters!
• $A \times B \ne B \times A$
• $A \times B = -(B \times A)$

c)
Properties of Cross Product
• Algebraic properties
• Distribution property
• Scalar multiplication property
• Etc.

• 1.
Using the Cross Product
If $u=<3, -2, 4>$ and $v=<-1, 2, -5>$, then compute $u \times v$.

• 2.
If $u= <0, 1, -3>$ and $v=<1, 0, 4>$, then compute $u \times v$.

• 3.
Showing Cross Product is Not Commutative Let $A= <1, 0, 3>$ and $B= <-2, 1, 0>$. Show that:

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

• 4.
Verifying Properties of Cross Product Suppose $c=2$, and $u=<2, -1, 4>$ and $v=<4, 2, 1>$. Verify that

$(cu) \times v = c(u \times v)$

• 5.
Suppose $u=<2, -1, 4>$, $v=<4, 2, 1>$ and $w=<1, -1, 2>$. Verify that

$u \times (v + w) = u \times v + u \times w$