Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Dot Product Take 3-D for example, De?nition If two vectors ?a = and ?b =, then the dot product (inner product, scalar product) of ?a and ?b is de?ned as ? a ? ? b = a 1 b 1 + a 2 b 2 + a 3 b 3 Example ? = ?14 (?i + 2?j ? 4?k) ? (?2?i +?j + 3?k) = ?12 Remark: Dot product gives a scalar. Properties: 1. ?a ??a = a2 1 + a2 2 + a2 3 = |?a|2 2. ?a ? ?b = a 1 b 1 + a 2 b 2 + a 3 b 3 = ?b ??a 3. ?a ? (?b + ?c) = ?a ? ?b +?a ? ?c 4. (c?a) ? ?b = c(a 1 b 1 + a 2 b 2 + a 3 b 3 ) = c(?a ? ?b) = ?a ? (c?b) 5. ?0 ??a = 0

# Dot Product (and Vector Projection)

Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Dot Product Take 3-D for example, De?nition If two vectors ?a = and ?b =, then the dot product (inner product, scalar product) of ?a and ?b is de?ned as ? a ? ? b = a 1 b 1 + a 2 b 2 + a 3 b 3 Example ? = ?14 (?i + 2?j ? 4?k) ? (?2?i +?j + 3?k) = ?12 Remark: Dot product gives a scalar. Properties: 1. ?a ??a = a2 1 + a2 2 + a2 3 = |?a|2 2. ?a ? ?b = a 1 b 1 + a 2 b 2 + a 3 b 3 = ?b ??a 3. ?a ? (?b + ?c) = ?a ? ?b +?a ? ?c 4. (c?a) ? ?b = c(a 1 b 1 + a 2 b 2 + a 3 b 3 ) = c(?a ? ?b) = ?a ? (c?b) 5. ?0 ??a = 0

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