1. Chapter 12 – Vectors and the Geometry of Space 12.4 The Cross Product 1

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. 12.4 The Cross Product2

3. Cross Product as Determinants To make Definition one easier, we will use the notation of determinants. A determinant of order 2 is defined by 12.4 The Cross Product3

4. Cross Product as Determinants A determinant of order 3 is defined in terms of second order determinates as shown below. 12.4 The Cross Product4

5. Cross Product as Determinants 12.4 The Cross Product5

6. Example 1 – pg. 814 #5 Find the cross product a x b and verify that it is orthogonal to both a and b. 12.4 The Cross Product6

7. 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 180 o from a to b, then your thumb points in the direction of axb. 12.4 The Cross Product7

8. Visualization The Cross Product 12.4 The Cross Product8

9. Theorems 9

10. Example 2 For 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) 12.4 The Cross Product10

11. 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) 12.4 The Cross Product11

12. Definition – Triple Products The 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: 12.4 The Cross Product12

13. Definition – Volume of a Parallelepiped 12.4 The Cross Product13

14. Example 3 – pg. 815 # 36 Find 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) 12.4 The Cross Product14

15. 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  12.4 The Cross Product15

16. Example 4 – pg. 815 #41 A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction at the end of the wrench. Find the magnitude of the force needed to supply 100 N  m of torque to the bolt. 12.4 The Cross Product16

17. 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 1 ◦ Example 2 Example 2 ◦ Example 5 Example 5 12.4 The Cross Product17

18. Demonstrations Feel free to explore these demonstrations below. Cross Product of Vectors 12.4 The Cross Product18