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03-cross - Lecture 3 Determinants and cross product...

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Lecture 3: Determinants and cross product http://www.math.columbia.edu/~dpt/F10/CalcIII/ September 14, 2010
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Announcements I Hand in homework. I Milbank help room is open 10AM–6PM Mon–Thurs, 11AM–3PM Fri. I Extra problems on back of the homework. I Next office hours: Wednesday, 2–3 PM.
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Lecture 3: Determinants and cross product I Dot product reprise Determinants Cross products
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Projections: Geometric description The dot product of ~ v and ~ w is ~ v · ~ w = v 1 w 1 + v 2 w 2 + v 3 w 3 = k ~ v kk ~ w k cos ( θ ) . The projection of ~ w onto ~ v , written proj ~ v ( ~ w ) , is the closest point to ~ w on the line containing ~ v .
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Projections: Geometric description The dot product of ~ v and ~ w is ~ v · ~ w = v 1 w 1 + v 2 w 2 + v 3 w 3 = k ~ v kk ~ w k cos ( θ ) . The projection of ~ w onto ~ v , written proj ~ v ( ~ w ) , is the closest point to ~ w on the line containing ~ v .
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Projections: Geometric description The dot product of ~ v and ~ w is ~ v · ~ w = v 1 w 1 + v 2 w 2 + v 3 w 3 = k ~ v kk ~ w k cos ( θ ) . The projection of ~ w onto ~ v , written proj ~ v ( ~ w ) , is the closest point to ~ w on the line containing ~ v .
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Projections and dot product proj ~ v ( ~ w ) is the vector in dir. of ~ v so ~ w - proj ~ v ( ~ w ) is perp. to ~ v . proj ~ v ( ~ w ) = a ~ v ( a to be determined) Solve for a : Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . (Note: Scalar times a vector.)
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Projections and dot product proj ~ v ( ~ w ) is the vector in dir. of ~ v so ~ w - proj ~ v ( ~ w ) is perp. to ~ v . proj ~ v ( ~ w ) = a ~ v ( a to be determined) Solve for a : Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . (Note: Scalar times a vector.)
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Projections and dot product proj ~ v ( ~ w ) is the vector in dir. of ~ v so ~ w - proj ~ v ( ~ w ) is perp. to ~ v . proj ~ v ( ~ w ) = a ~ v ( a to be determined) Solve for a : Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . (Note: Scalar times a vector.)
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Projections and dot product proj ~ v ( ~ w ) is the vector in dir. of ~ v so ~ w - proj ~ v ( ~ w ) is perp. to ~ v . proj ~ v ( ~ w ) = a ~ v ( a to be determined) Solve for a : Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . (Note: Scalar times a vector.)
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Projections and dot product, cont Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . The (signed) length of the projection is comp ~ v ( ~ w ) = ±k proj ~ v ( ~ w ) k = ~ w · ~ v k ~ v k ~ v · ~ w = k ~ v k comp ~ v ( ~ w ) = comp ~ w ( ~ v ) k ~ w k
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Lecture 3: Determinants and cross product Dot product reprise I Determinants Cross products
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Matrices and determinants Before getting to the cross product, we’ll look at determinants .
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