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Chapter 12 – Vectors and the Geometry of Space12.4 The Cross Product

12.4 The Cross Product

12.4 The Cross Product 2

Definition – Cross Product

Note: The result is a vector. Sometimes the cross product is called a vector product. This only works for three dimensional vectors.

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Cross Product as Determinants

To make Definition one easier, we will use the notation of determinants. A determinant of order 2 is defined by

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Cross Product as Determinants

A determinant of order 3 is defined in terms of second order determinates as shown below.

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Cross Product as Determinants

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Example 1 – pg. 814 #5Find the cross product a x b and

verify that it is orthogonal to both a and b.

1 1

2 2

a i j k

b i j k

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Theorem 5

The direction of axb is given by the right hand rule: If your fingers of your right hand curl in the direction of a rotation of an angle less than 180o from a to b, then your thumb points in the direction of axb.

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VisualizationThe Cross Product

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Theorems

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Example 2For the below problem, find the

following:◦a nonzero vector orthogonal to the

plane through the points P, Q, and R.◦the area of triangle PQR.

P(2,1,5) Q(-1,3,4)R(3,0,6)

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Theorem 8

Note: The cross product is not commutative

i x j j x i

Associative law for multiplication does not hold.

(a x b) x c a x (b x c)

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Definition – Triple ProductsThe product a (b x c) is called

the scalar triple product of vectors a, b, and c. We can write the scalar triple product as a determinant:

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Definition – Volume of a Parallelepiped

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Example 3 – pg. 815 # 36Find the volume of the

parallelepiped with adjacent edges PQ, PR, and PS.

P(3,0,1) Q(-1,2,5)R(5,1,-1) S(0,4,2)

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Torque Cross product occurs often in physics. Let’s consider a force, F, acting on a rigid body at a

point given by a position vector r. (i.e. tightening a bolt by applying force to a wrench). The torque is defined as

= r x F

and measures the tendency of the body to rotate about the origin.

The magnitude of the torque vector is

||= |r x F| = |r||F|sin

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Example 4 – pg. 815 #41A wrench 30 cm long lies along

the positive y-axis and grips a bolt at the origin. A force is applied in the direction <0,3,-4> at the end of the wrench. Find the magnitude of the force needed to supply 100 Nm of torque to the bolt.

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More Examples

The video examples below are from section 12.4 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 1◦Example 2◦Example 5

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Demonstrations

Feel free to explore these demonstrations below.

Cross Product of Vectors