The Cross Product of 2 Vectors

11.3

JMerrill, 2010

Unit Vectors in 2D

• In 2-D space, the unit vectors <0,1> and <1,0> are the standard unit vectors and denoted by i = <1,0> and j = <0,1>

j = <0,1>

i = <1,0>

Unit Vectors in 2D

• Any vector can be written as a linear combination of the vectors I and j.

• v = <v1, v2>

= v1<1,0> + v2<0,1>

= v1i + v2j

• The scalars v1 and v2 are the horizontal and vertical components of v.

Writing a Linear Combination of Unit Vectors

• u has initial point (2, -5) and terminal point (-1,3). Write u as a linear combination of the unit vectors i & j.

• u = <-1-2, 3-(-5)> = <-3, 8> = -3i + 8j

Unit Vectors in 3D

The Cross Product

Finding The Cross Product

• An easy way to calculate the cross product is to use a matrix. We use the determinant form with cofactor expansion.

Finding The Cross Product

• An easy way to calculate the cross product is to use a matrix. We use the determinant form with cofactor expansion.

Finding the Cross Product

Subtraction sign Addition Sign

Example

• Given u = i + 2j + k and v = 3i + j + 2k, find the cross product of u x v.

1 2 1

3 1 2

i j k

u x v 2 1 1 1 1 2

1 2 3 2 3 1

i j k

4 1 2 3 1 6( ) ( ) ( )i j k

3 5i j k

You Try

• Given u = i + 2j + k and v = 3i + j + 2k, find the cross product of v x u.

3 1 2

1 2 1

i j k

v x u 1 2 3 2 3 1

2 1 1 1 1 2

i j k

1 4 3 2 6 1( ) ( ) ( )i j k

3 5i j k

Using the Cross Product

• Find a unit vector that is orthogonal to both u = 3i – 4j + k and v = -3i + 6j.

• The cross product gives a vector that is orthogonal to both u and v

= -6i – 3j + 6k

• The question asks for a unit vector that’s orthogonal.

3 4 1

3 6 0

-

-

i j k

u x v

Using the Cross Product

• So, we need to divide by the magnitude of the orthogonal vector.

• -6i – 3j + 6k2 2 26 3 6 81 9 ( ) ( )u x v

2 1 2

3 3 3

u x vi j k

u x v

Triple Scalar Product

• Given 3 vectors u = 3i – 5j + k v = 2j – 2k w = 3i + j + k

• Find the volume of a parallelepiped having these vectors as adjacent edges.

• The volume is found by V = |u∙(v x w)|

Triple Scalar Product

3 5 1

0 2 2

3 1 1

-

( ) -

u v x w

2 2 0 2 0 23 5 11 1 3 1 3 1

- - ( )

3 4 5 6 1 6( ) ( ) ( )

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