Engineering Calculus Notes 63

Engineering Calculus Notes 63 - | v | and | w | are both...

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1.4. PROJECTION OF VECTORS; DOT PRODUCTS 51 Proposition 1.4.3. The dot product has the following geometric properties: 1. −→ v · −→ w = | −→ v || −→ w | cos θ , where θ is the angle between the “arrows” representing −→ v and −→ w . 2. −→ v · −→ w = 0 precisely if the arrows representing −→ v and −→ w are perpendicular to each other, or if one of the vectors is the zero vector. 3. −→ v · −→ v = | −→ v | 2 4. proj −→ w −→ v = p −→ v · −→ w −→ w · −→ w P −→ w (provided −→ w n = −→ 0 ). We note the curiosity in the second item: the dot product of the zero vector with any vector is zero. While the zero vector has no well-deFned direction, we will Fnd it a convenient Fction to say that the zero vector is perpendicular to every vector, including itself . Proof. 1. This is just Equation ( 1.17 ). 2. This is an (almost) immediate consequence: if
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Unformatted text preview: | v | and | w | are both nonzero ( i.e. , v n = n = w ) then v w = 0 precisely when cos = 0, and this is the same as saying that v is perpendicular to w (denoted- v - w ). But if either v or w is 0 , then clearly v w = 0 by either side of Equation ( 1.17 ). 3. This is just (1) when v = w , which in particular means = 0, or cos = 1. 4. This follows from Equation ( 1.15 ) by substitution: proj w v = p | v | | w | cos P w = | v || w | | w | 2 cos w = p v w w w P w....
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