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13.3-4 Products

13.3-4 Products - ET 12.3 M 13.3 Dot Product Arithmetic...

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ET 12.3; M 13.3. Dot Product Arithmetic definition. In R 3 , the scalar product of vectors ~a and ~ b is ~a · ~ b = a 1 b 1 + a 2 b 2 + a 3 b 3 . Connections with geometry. For lengths, ~a · ~a = | ~a | 2 . On directions, two nonzero vectors are perpen- dicular if and only if their dot product is equal to 0. When ~a 6 = ~ 0, the vector projection proj ~ a ~ b of ~ b onto ~a has the form s~a , where s = ( ~ b · ~a ) ( ~a · ~a ) = ~ b · ~a | ~a | 2 . Also, s~a = | ~ b | (cos θ a = | ~ b | (cos θ ) ~a | ~a | , where θ is the smallest nonegative angle between the vectors ~ b and ~a . It follows that ~ b · ~a = | ~ b || ~a | cos θ, and that cos θ = ~ b · ~a | ~ b || ~a | . 1
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ET 12.4; M 13.4. Cross Product Special fact in 3 dimensions. If ~a and ~ b are not aligned, then every vector that is perpendicular to both ~a and ~ b is a multiple of the vector ( a 2 b 3 - a 3 b 2 ) ˆ i + ( a 3 b 1 - a 1 b 3 ) ˆ j + ( a 1 b 2 - a 2 b 1 ) ˆ k. This vector product ~a × ~ b can also be computed in other ways. Magnitude and direction. The length of ~a × ~ b is | ~a || ~ b | sin θ, for the same angle
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