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EXAMPLE 5 Use the scalar triple product to show that the vectors a b 2, 1, 4 , and c 0, 1, 4, 7, 9, 18 are coplanar. SOLUTION We use Equation 10 to compute their scalar triple product: a b 1
0 c 1 4
9 1 18 7
0 4 4 36 7 4
18 7 2
9 0 Therefore, by (11) the volume of the parallelepiped determined by a, b, and c is 0. This
means that a, b, and c are coplanar. r
¨ The idea of a cross product occurs often in physics. In particular, we consider a force F
acting on a rigid body at a point given by a position vector r. (For instance, if we tighten
a bolt by applying a force to a wrench as in Figure 4, we produce a turning effect.) The
torque (relative to the origin) is deﬁned to be the cross product of the position and force
FIGURE 4 r F and measures the tendency of the body to rotate about the origin. The direction of the
torque vector indicates the axis of rotation. According to Theorem 6, the magnitude of the
torque vector is
r F r F sin where is the angle between the position and force vectors. Observe that the only component of F that can cause a rotation is the one perpendicular to r, that is, F sin . The
magnitude of the torque is equal to the area of the parallelogram determined by r and F. 5E-13(pp 848-857) 856 ❙❙❙❙ 1/18/06 11:22 AM Page 856 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 6 A bolt is tightened by applying a 40-N force to a 0.25-m wrench as shown in
Figure 5. Find the magnitude of the torque about the center of the bolt.
0.25 m SOLUTION The magnitude of the torque vector is 40 N r F r 10 sin 75 F sin 75 9.66 N m 0.25 40 sin 75
9.66 J If the bolt is right-threaded, then the torque vector itself is
n F IGURE 5 9.66 n where n is a unit vector directed down into the page. |||| 13.4 Exercises 1–7 |||| Find the cross product a
to both a and b. 1. a 1, 2, 0 , b 2. a 5, 1, 4 , b 3. a 2i 4. a i 5. a 3i 6. a i k, j k, 1, 0, 2
b ■ j b 2k b i j k b i 2j e t k, et j ■ z 4...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
- Fall '09