Cross product

Intro Lesson: a
13:34
Intro Lesson: b
11:27
Intro Lesson: c
4:01
Lesson: 1
5:55
Lesson: 2
4:52
Lesson: 3
7:47
Lesson: 4
10:32
Lesson: 5
11:46

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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:

$u \times (v+w) = u \times v + u \times w$
$(u + v) \times w = u \times w + v \times w$
$(cu) \times v = c(u \times v) = u \times (cv)$
$u \cdot (v \times w ) = (u \times v) \cdot w$
$u \times (v \times w ) = (u \cdot w) v - (u \cdot v) w$

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

Cross product

Cross product

Lessons

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