13.04b Triple Products & Torque The product a   b  c  is called the scalar triple product. (Notice that because it contains a dot product, the result is a scalar.) We can write the scalar triple product as a determinant: a1 a   b  c   b1 c1

a2 b2 c2

a3 b3 c3

Consider a parallelepiped determined by vectors a , b , and c . (A parallelepiped looks like a box that has been pushed over so that the corner angles are not 90o.) The area of the base is: A  b  c If  is the angle between a and b  c , then h  a cos  .

h

 a

The volume of the parallelepiped is: c

V  Ah  b  c a cos   a   b  c 

b

The volume of a parallelepiped determined by vectors a , b , and c is the magnitude of their scalar triple product: V  a  b  c

If the vectors a , b , and c lie in the same plane, then the parallelepiped would have a volume of zero. This leads us to the following fact: If the scalar triple product of three vectors is zero, then the vectors are coplanar.

Hanford High School Calculus III

13.04b Triple Products & Torque – G. Kelly

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Ex. 5: Use the scalar triple product to show that the vectors a  1, 4, 7 , b  2, 1, 4 , and c  0, 9,18 are coplanar.

1 4 7 1 4 2 4 2 1 a  b  c  2 1 4  1 4 7 9 18 0 18 0 9 0 9 18

 1 18   36    4  36  0   7  18  0   118  4  36   7  18  18  144  126

0

Torque – an application from physics If a force F is applied to a rigid body at a point given by position vector r , we produce a turning or twisting effect. (An example would be a force applied to the end of a wrench turning a bolt.) This twisting force is called the torque τ .

τ

r

F 

The torque is defined to be the cross product of the position and force vectors:

τ  rF The direction of the torque vector indicates the axis of rotation. The magnitude of the torque vector is: τ  r  F  r F sin  where  is the angle between the position vector and the force vector. Note: F sin  is the component of the force vector that is perpendicular to the position vector. In real life, the torque is usually calculated simply by multiplying force times distance. Any force that is not applied perpendicular to the wrench is just wasted effort.

Hanford High School Calculus III

13.04b Triple Products & Torque – G. Kelly

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Ex. 6: A bolt is tightened by applying a 40-N force to a 0.25 m wrench as shown. Find the magnitude of the torque about the center of the bolt.

0.25 m 40 N

75o The magnitude of the torque vector is: τ  r  F  r F sin 75o 

 0.25 40  sin 75o

 10sin 75o  9.66 N  m

Hanford High School Calculus III

13.04b Triple Products & Torque – G. Kelly

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