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Unformatted text preview: Math 20C Multivariable Calculus Lecture 4 1 Slide 1 ’ & $ % Cross product and determinants (Sec. 12.4) • Review: The dot product is a number. (Sec. 12.3) • Geometric definition of cross product. • Properties, and determinants. • Cross product in components. • Triple product and volumes. Slide 2 ’ & $ % There are two main ways to introduce the cross product Geometrical definition → Properties → Expression in components. Geometrical expression ← Properties ← Definition in components. We choose the first way. The book chooses the second way Math 20C Multivariable Calculus Lecture 4 2 Slide 3 ’ & $ % The cross product of two vectors is another vector Definition 1 Let v , w be 3dimensional vectors, and ≤ θ ≤ π be the angle in between them. Then, v × w is a vector perpendicular to v and w , pointing in the direction given by the right hand rule, and with norm  v × w  =  v  w  sin( θ ) . The cross product is perpendicular to the original vectors Slide 4 ’ & $ % The direction of the cross product vector is given by the right hand rule V W W x V V x W O Tails together. Math 20C Multivariable Calculus Lecture 4 3 Slide 5 ’ & $ % The cross products of the vectors i, j and k are simple to compute i × j = k , j × i = k , j × k = i , k × j = i , k × i = j , i × k = j . i j k x y z Slide 6 ’ & $ % Main properties of the cross product • v × w = w × v ⇒ v × v = 0, • ( a v ) × w = v × ( a w ) = a ( v × w ), • u × ( v + w ) = u × v + u × w , • u × ( v × w ) = ( u · w ) v ( u · v ) w . Math 20C Multivariable Calculus Lecture 4 4 Slide 7 ’ & $ % The cross product is not associative That is, u × ( v × w ) 6 = ( u × v ) × w . Example: i × ( i × k ) = k ; but ( i × i ) × k = 0. The cross product of two vectors vanishes when the vectors are parallel Theorem 1 v , w 6 = 0 and v × w = 0 ⇔ v parallel w . Slide 8 ’ & $ % The length of a cross product vector is an area Theorem 2  v × w  is the area of the parallelogram formed by v and w . V W V sin(O) O Math 20C Multivariable Calculus Lecture 4 5 Slide 9 ’ & $ % The cross product can be written in terms of the components of the original vectors Theorem 3 Let v = h v 1 , v 2 , v 3 i , and w = h w 1 , w 2 , w 3 i . Then, v × w = h ( v 2 w 3 v 3 w 2 ) , ( v 3 w 1 v 1 w 3 ) , ( v 1 w 2 v 2 w 1 ) i . For the proof of the last theorem, recall that i × j = k , j × k = i , k × i = j . Slide 10 ’ & $ % We recall here the definition of determinant of a matrix We use determinants only as a tool to remember the components of v × w . ¯ ¯ ¯ ¯ ¯ a b c d ¯ ¯ ¯ ¯ ¯ = ad bc. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = a 1 ¯ ¯ ¯ ¯ ¯ b 2 b 3 c 2 c 3 ¯ ¯ ¯ ¯ ¯ a 2 ¯ ¯ ¯ ¯ ¯ b 1 b 3 c 1 c 3 ¯ ¯ ¯ ¯ ¯ + a 3 ¯ ¯ ¯ ¯ ¯ b 1 b 2 c 1 c 2 ¯ ¯ ¯ ¯ ¯ Math 20C Multivariable Calculus Lecture 4 6 Slide 11 ’ & $ % The triple product of three vectors is a number Definition 2...
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This note was uploaded on 04/30/2008 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.
 Spring '08
 Helton
 Determinant, Multivariable Calculus, Dot Product

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