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HH_13_4 - P = 1 1 1 and adjacent vertices Q = 4 4 R = 5 7 6...

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(10/14/08) Math 10C. Lecture Examples. Section 13.4. The cross product Example 1 Evaluate the determinant, vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 3 2 4 - 1 0 6 5 1 - 2 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle . Answer: The given determinant equals 34. Example 2 Find the cross product of v = ( 3 , 1 , - 2 ) and w = ( 0 , 4 , 2 ) . Answer: v × w = ( 10 , - 6 , 12 ) Example 3 As a partial check of the result of Example 2, show that each the given vectors is perpendicular to the calculated cross product. Answer: Let u = ( 10 , - 6 , 12 ) be the calculated cross product. v · u = 0 w · u = 0 Example 4 Find a nonzero vector perpendicular to v = 4i - j + k and w = 2i - k . Answer: One answer: The cross product v × w = i + 6 j + 2 k is perpendicular to v and w . Example 5 Find the area of the triangle with vertices P = ( 1 , 2 , 3 ) , Q = ( 4 , 2 , 6 ) and R = ( 5 , 3 , 7 ) . Answer: [Area of the triangle] = 3 2 2 Example 6 Calculate the scalar triple product, u · ( v × w ) for u = ( 3 , 3 , - 1 ) , v = ( 4 , 6 , 5 ) , and w = ( 2 , 2 , - 1 ) . Answer: u · ( v × w ) = - 2 Parallelepiped Tetrahedron
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Unformatted text preview: P = ( 1 , 1 , 1 ) and adjacent vertices Q = ( 4 , 4 , ) , R = ( 5 , 7 , 6 ) , and S = ( 3 , 3 , ) ? Answer: [Volume of the parallelepiped] = 2 Example 8 The vectors i , j , and k with their bases at the origin form three edges of a tetrahedron. What is its volume? Answer: [Volume] = 1 6 Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/: ‡ Section 12.4: Examples 1–8 † Lecture notes to accompany Section 13.4 of Calculus by Hughes-Hallett. ‡ The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. 1...
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