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

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Unformatted text preview: ET 12.3; M 13.3. Dot Product Arithmetic definition. In R3 , the scalar product of vectors a and b is a · b = a1 b1 + a2 b2 + a3 b3 . Connections with geometry. For lengths, a · a = |a|2 . On directions, two nonzero vectors are perpendicular if and only if their dot product is equal to 0. When a = 0, the vector projection proja b of b onto a has the form sa, where s= Also, (b · a) b·a = . 2 (a · a) |a| a sa = |b|(cos θ)ˆ = |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 1 cos θ = b·a |b||a| . 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 ˆ (a2 b3 − a3 b2 )ˆ + (a3 b1 − a1 b3 )ˆ + (a1 b2 − a2 b1 )k. i j 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 θ as before. When a and b are not aligned, there are two vectors perpendicular to both a and b and having the length displayed above. Then a × b is the one that makes the triad {a, b, c} right-handed. This distinction between orientations matters in physical applications. Algebraic facts. Products of nonzero vectors can be equal to zero. Both products distribute with addition, subtraction and rescaling. The dot product is commutative, while the cross product is anticommutative. Neither product is associative. But a · (b × c) = (a × b) · c. Both sides here give signed volumes of parallepipeds. The End 12 January 2007 ...
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This note was uploaded on 01/26/2011 for the course MATH 200 taught by Professor Unknown during the Spring '03 term at The University of British Columbia.

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

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