The Cross Product of Two Vectors In Space

Section 10.32015

Precalculus Extra Credit Proj.

• Due 4/30

Just as 2 points in space determine a vector, 2 vectors in space determine a plane.

Many applications in physics and engineering involve finding a vector in space that is orthogonal to two given vectors.

In this lesson you will learn:

•How to find the cross product of 2 vectors in space.

•Applications of the cross product.

•How to find the triple scalar product.

•Applications of the triple scalar product.

The Cross Product: A vector in space that is orthogonal to two given vectors. (Note: that while the dot product was a scalar, the cross product is a vector.)

The cross product of u and v is the vector u x v.

The cross product of two vectors, unlike the dot product, represents a vector.

A convenient way to find u x v is to use a determinant involving vector u and vector v.

1 2 3

1 2 3

i j k

u v u u u

v v v

The cross productis found by taking this determinant.

2,4,5 1, 2, 1u and v

2 4 5 ( 4 5 4 ) (4 10 2 )

1 2

6 7 8

1

i j k

i j k k i j kj i

Find the cross product for the vectors below, then find the magnitude of the cross product.

u x v and v x u have equal lengths in opposite directions. Both u x v and v x u are perpendicular to the plane determined by u and v.

149u v

2,4,5 , 1, 2, 1 6,7, 8u v and u v

Let’s look at the 3 vectors from the last problem

What is the dot product of

2,4,5 6,7, 8u with u v

And

1, 2, 1 6,7, 8v with u v ?

If you answered 0 in both cases, you would be correct.Recall that whenever two non-zero vectors are perpendicular, their dot product is 0. Thus the cross product creates a vector perpendicular to the vectors u and v.

0

0

?

1.

2.

3.

4. 0 0 0

5. 0

6.

u v v u

u v w u v v w

c u v cu v u cv

u u

u u

u v w u v w

Algebraic Properties of the Cross Product:

Example, You try:

1. Find a unit vector that is orthogonal to both :

3 4 3 6u i j k and v i j

2 1 2

3 3 3i j k

u

v

Area of a parallelogram = bh, in this diagram, area=

h

sin ,h u

.v h

Since sinarea v u

2 vectors in space form a parallelogram

sin ,h

u

sinu v v u

A geometric property of the cross product is:

1. is orthogonal to both u and v

2. area of parallelogram having u and v as adjacent sides

3. sin

4. 0 if and only if u and v are scalar multiples of ea. other

15. area of a triangle having u a

2

u v

u v

u v u v

u v

u v

nd v as sides

Geometric Properties of the Cross Product:

Geometric application example. You try:

Find the area of the triangle with the given vertices.

The area A of the triangle having u and v as adjacent sides is given by:

To begin, create 2 vectors representing adjacent sides of the triangle. Make sure they have the same initial point.

1

2A u v

A(1,-4,3) B(2,0,2) C(-2,2,0)

1396

2

136 11

2 3 11 .sq units

1area of triangle having u and v as sides

2u v

Geometric application example:

Show that the quadrilateral with vertices at the followingpoints is a parallelogram. Find the area of the parallelogram.Is the parallelogram a rectangle?

A(5,2,0) B(2,6,1) C(2,4,7) D(5,0,6)

To begin, plot the vertices below, then find the 4 vectors representing the sides of the Parallelogram, and use the property:

area of parallelogram having u and v as adjacent sidesu v

Show that the quadrilateral with vertices at the following points is a parallelogram. Find the area of the parallelogram. Is the parallelogram a rectangle? A(5,2,0) B(2,6,1) C(2,4,7) D(5,0,6)

x

y

zAB BBBBBBBBBBBBBB

CD BBBBBBBBBBBBBB

AD BBBBBBBBBBBBBB

CB BBBBBBBBBBBBBB

AB AD BBBBBBBBBBBBBBBBBBBBBBBBBBBB

Is the parallelogram a rectangle?

3,4,1

3, 4, 1

0, 2,6

0,2, 6

32.19

no

Triple Scalar Product: For the vectors u, v, and w in space, the dot product of u and is called the triple scalar product of u, v, and w.

A Geometric property of the triple scalar product:

The volume V of a parallelepiped with vectors u, v, and w as adjacent edges is given by:

1 2 3

1 2 3

1 2 3

u u u

u v w v v v

w w w

v u v w

A parallelepiped is a figure created when a parallelogram has depth

v w

Example. You Try:

1. Find the volume of a parallelepiped having adjacent edges:

3 5 2 2 3u i j k v j k w i j k

36

Homework:

Day 1: Pg. 726 5-43 odd

Homework:

Day 2: Pg. 726 6-44 even

A note about the differing angles between 2 vectors depending on which formula is used:

In the quadrilateral example, when the sine formula is used to find the angle between the vectors, , yet when the cosine formula is used, the

The angle found with the cosine formula is 180-the other angle.

Using a positive value for the dot product in the cosine formula yields the same angle.

So they either find the same angle, or the angle of one vector with the opposite of the other vector.

86.4 93.6

The dot product is the scalar product. Scalar product means

the product value is the scalar quantity which is obtained by

the cosine angle between two vectors and the cross product

means the vector product which implies that the product

value is vector quantity. The vector quantity is obtained by

the sine angle between the two vectors

Cos theta vs. sine theta