The Dot Product DEFINITION 13.3.1 For vectors a=ai+azj+as k and b=bii+b2j+b3 k. we define the dot product a.b by setting a·b=a1b1+a2b2+a3b3. (13.3.2) a…a=lal2. (13.3.7) a·b=lalllbll cos0. Main Meny own6
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The Cross Product DEFINITION 13.4.1 If a and b are not parallel,then a x bis the vector with the following properties: 1.a x b is prependicular to the plane of a and b. 2.a,b,a x b form a right-handed triple 3.lla x bll =llallbll sin where 6 is the angle between a and b. If a and b are parallel,then a x b=0. a×b b×a=-(a×b). Figure 13.4.1 Main Meny 5
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The Cross Product THEOREM 13.4.9 For vectors a ai+aj+a3k and b=bii+b2j+b3k. a x b=(a2b3-a3b2)i-(a1b3-a3b1)j+(a1b2-a2b)k THEOREM 13.4.9 For vectors a =ai+aj+ask and b=bi+b2j+b3k, i j k a x b= Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Cross Product
The Cross Product The Scalar Triple Product (13.4.6) (a x b).cl. Figure 13.4.6 (13.4.7八 (a×b)c=(c×a)b=(b×c)a. Main Meny own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Cross Product The Scalar Triple Product
(a×b)c= b b2 b3 C2 Main Menu☐ 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Verify a×+(a.b)2=l25 Main Menu 0
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. 2 2 2 2 a b (a b) a b Verify + =
ax(bxc)=(a.c)b-(a.b)c (axb)xc=(c-a)b-(c.b)a Main Meny☐ o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. a b c a c b a b c ( ) = ( ) − ( ) a b c c a b c b a ( ) = ( ) − ( )