10.4 Cross product: a vector orthogonal to two given vectors

Cross product of two vectors in space (area of parallelogram)

Triple Scalar product (volume of parallelepiped)

Torque

Cofactor Expansion

2nd method (not in book)

The Cross Product:Many applications in physics and engineering involve finding a vector in space that is orthogonal to two given vectors. This vector is called the Cross product. (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.

• The cross product can be expressed as

• Expanding the determinants gives

Vector Products Using Determinants

Find the cross product for the vectors below. u = <2, 4, 5> and v = <1, -2, -1>

Now that you can do a cross product the next step is to see why this is useful.

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 ?

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.

?

Example, You try:

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

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

Vector Products of Unit Vectors

Contrast with scalar products of unit vectors

Signs are interchangeable in cross products

If A & B are vectors, their Vector (Cross) Product is defined as:

A third vector

The magnitude of vector C is AB sinθ where θ is the angle between A & B

• The magnitude of C, which is |A||B| sinθ or |AxB| is equal to the area of the parallelogram formed by A and B.

• The direction of C is perpendicular to the plane formed by A and B

• The best way to determine this direction is to use the right-hand rule

Magnitude of Cross Product

u

v

Area of a parallelogram = bh, in this diagram,

h

sin ,h u

v h

Since sinarea v u

2 vectors in space form a parallelogram

Calculate the area of the trianglewhere P = (2, 4, -7), Q = (3, 7, 18), and R = (-5, 12, 8)

Now you try! Find the area of the triangle with the given verticesA(1, -4, 3) B(2, 0, 2) C(-2, 2, 0)

Calculate the area of the parallelogram PQRS, where P = (1, 1), Q = (2, 3), R = (5, 4), and S = (4, 2)

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, then find the 4 vectors representing the sides of the Parallelogram, and use the property:

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

z

Is the parallelogram a rectangle?

Triple Scalar Product or Box 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 w

v u v w

A parallelepiped is a figure created when a parallelogram has depth

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

Torque (moment of force)• Tendency of force to rotate an object about an axis, fulcrum, or

pivot.• “twist” of an object• Greek “tau”• Equation where F is force and P is the pivot point.

T=PQ x FWhere magnitude of T measures the tendency of PQ to rotate

counterclockwise about axis directed along T. Another way to look at this is:

Magnitude of torque vector=|r||F|sin x or |r x F| where r is the length of the lever arm and the scalar component of F is

perpendicular to r.

Pg. 748 Torque problemVertical force of 50 pounds applied to a 1-ft lever attached to an axle at P. Find the moment of force about P when θ=60.

5-11 ODD, 31-37 ODD, 41, 42, 45, 46, 47

Homework/Classwork

TI 89 Graphing Calculator