Lecture 3 - Dot Product Scalar Product Dot Product Scalar...

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ot Product / Scalar Product Dot Product / Scalar Product Can be thought of as a measure of the “right-angularity” of two vectors θ cos V U V U v v v v = Definition, In terms of components, ˆ ˆ ˆ v ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ + + = + + = k V j V i V V k U j U i U U z y x z y x v v v 10 0 ... ) ( ) ( ) ( ) ( + + + + = i j V U k i V U j i V U i i V U V U x y z x y x x x z z y y x x V U V U V U V U + + = v v So,
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arallel Component Parallel Component θ cos U U p v v =
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arallel Component Parallel Component θ cos ˆ ˆ U e U e v v = Since the mag. of a unit vector = 1, ˆ v v And we previously determined, cos U U e = Therefore, cos U U p v v = U e U p ˆ ˆ ˆ v v v v = ( ) e U e U p
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ormal Component Normal Component v v v Knowing the Triangle Rule, n p U U U + Therefore, p n U U U v v v =
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ross Product Cross Product Very useful for determining moments otice that result is a vector!!! Definition, ˆ n v v v v Notice that result is a vector!!! What if 0 o (Parallel Lines)? ( ) e V U V U sin θ = ×
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ross Product Cross Product NOT Commutative, U V V U v v v v × × U V V U v v v v v v v v v v × = × When multiplying by a scalar, Distributive: ( ) ( ) ( ) () ( ) W U V U W V U V a U V U a V U a v v v v v v v × + × = + × × = × = ×
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ross Product: In terms of components k j i ) ) ) Cross Product: In terms of components z y x z y x V V V U U U V U v v = × j i k j i ) ) )
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