Cross product examples

# Multivariable calculus and vector analysis

## A set of on-line readings

### Non-Math ML version

### Cross product examples

#### Example 1

Calculate the cross product between a = (3,-3, 1) and b = (4, 9, 2).

Solution: The cross product is

a × b | = |  |
| = i(-3 ⋅ 2 - 1 ⋅ 9) - j(3 ⋅ 2 - 1 ⋅ 4) + k(3 ⋅ 9 + 3 ⋅ 4) |  |
| = -15i - 2j + 39k |  |

#### Example 2

Calculate the area of the parallelogram spanned by the vectors a = (3,-3, 1) and b = (4, 9, 2).

Solution: The area is ∥a × b∥. Using the above expression for the cross product, we find that the area is = 5.

#### Example 3

Calculate the area of the parallelogram spanned by the vectors a = (3,-3, 1) and c = (-12, 12,-4).

Solution:

a × c | = |  |
| = (0, 0, 0) |  |

The magnitude of the zero vector is zero, so the area of the parallelogram is zero. What happened?

## Contents by lecture

### Part 1

- Introduction
- Quiz 1: Practice Quiz (not graded)
- Interactive concept-visualization tools
- Parametrization of a line
- Matrices and determinants
- The cross product
- The component formula for the cross product
- The scalar triple product
- The relationship between determinants and area or volume
- Parametrization of a line examples
- Cross product examples
- Scalar triple product example

### Part 2

### Part 3

### Part 4

### Part 5

### Part 6

### Part 7

### Part 8

### Part 9

### Part 10

### Part 11

### Part 12

### Part 13

### Part 14

### Part 15

### Part 16

### Part 17

### Part 18

### Part 19

### Part 20

### Part 21

### Part 22

### Part 23

### Part 24

### Part 25

### Part 26

### Part 27