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lect10_3

# lect10_3 - The Dot Product(10.3 1 Dot Product u v The dot...

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The Dot Product - (10.3) 1. Dot Product The dot product of two vectors u ! "# u 1 , u 2 , u 3 \$ and v ! "# v 1 , v 2 , v 3 \$ is defined by u ! ! v ! " u 1 v 1 % u 2 v 2 % u 3 v 3 . The dot product of two vectors u ! "# u 1 , u 2 \$ and v ! "# v 1 , v 2 \$ is defined by u ! ! v ! " u 1 v 1 % u 2 v 2 . Notice that the dot product of two vectors is a scalar (not a vector). Example Compute the dot product u ! ! v ! and v ! ! w ! where u ! "# ! 1,2,3 \$ , v ! "# 2, ! 1,1 \$ and w ! "# 3,2, ! 4 \$ . u ! ! v ! "# ! 1,2,3 \$ ! # 2, ! 1,1 \$" " ! 1 #" 2 # % " 2 #" ! 1 # % " 3 #" 1 # " ! 1 v ! ! w ! "# 2, ! 1,1 \$ ! # 3,2, ! 4 \$" 2 " 3 # % " ! 1 #" 2 # % " 1 #" ! 4 # " 0. Notice that v ! ! w ! " 0 but v ! " 0 ! and w ! " 0. 2. Properties of the Dot Product Let u ! , v ! and w ! be vectors and c be a constant. Then a. u ! ! v ! " v ! ! u ! b. u ! ! " v ! % w ! # " u ! ! v ! % u ! ! w ! c. " cu ! # ! v ! " c " u ! ! v ! # " u ! ! " cv ! # d. 0 ! ! u ! " 0 e. u ! ! u ! " || u ! || 2 u ! ! u ! " u 1 2 % u 2 2 % u 3 2 " || u ! || 2 3. The Angle Between Two Vectors Let u ! and v ! be two nonzero vectors. The angle between u ! and v ! in the plane obtained by u ! and v ! is defined by the smaller angle " between them where 0 # " # # . Observe that a. If two vectors are in the same direction, then " " 0. b. If two vectors are in an opposite direction, then " " # .

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lect10_3 - The Dot Product(10.3 1 Dot Product u v The dot...

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