261-S12-sg1

# 261-S12-sg1 - MA 261 Spring 2012 Study Guide 1 1 Vectors in...

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Unformatted text preview: MA 261 - Spring 2012 Study Guide # 1 1. Vectors in R 2 and R 3 (a) ⃗ v = ⟨ a, b, c ⟩ = a ⃗ i + b ⃗ j + c ⃗ k ; vector addition and subtraction geometrically using paral- lelograms spanned by ⃗ u and ⃗ v ; length or magnitude of ⃗ v = ⟨ a, b, c ⟩ , | ⃗ v | = √ a 2 + b 2 + c 2 ; directed vector from P ( x , y , z ) to P 1 ( x 1 , y 1 , z 1 ) given by ⃗ v = P P 1 = P 1 − P = ⟨ x 1 − x , y 1 − y , z 1 − z ⟩ . (b) Dot (or inner) product of ⃗ a = ⟨ a 1 , a 2 , a 3 ⟩ and ⃗ b = ⟨ b 1 , b 2 , b 3 ⟩ : ⃗ a · ⃗ b = a 1 b 1 + a 2 b 2 + a 3 b 3 ; properties of dot product; useful identity: ⃗ a · ⃗ a = | ⃗ a | 2 ; angle between two vectors ⃗ a and ⃗ b : cos θ = ⃗ a · ⃗ b | ⃗ a || ⃗ b | ; ⃗ a ⊥ ⃗ b if and only if ⃗ a · ⃗ b = 0; the vector in R 2 with length r with angle θ is ⃗ v = ⟨ r cos θ, r sin θ ⟩ : x y θ r (c) Projection of ⃗ b along ⃗ a : proj ⃗ a ⃗ b = { ⃗ a · ⃗ b | ⃗ a | } ⃗ a | ⃗ a | ; Work = ⃗ F · ⃗ D . b proj a proj a b b a a b (d) Cross product (only for vectors in R 3 ): ⃗ a × ⃗ b = ⃗ i ⃗ j ⃗ k a 1 a 2 a 3 b 1 b 2 b 3 = a 2 a 3 b 2 b 3 ⃗ i − a 1 a 3 b 1 b 3 ⃗ j + a 1 a 2 b 1 b 2 ⃗ k properties of cross products; ⃗ a × ⃗ b is perpendicular (orthogonal or normal) to both ⃗ a and ⃗ b ; area of parallelogram spanned by ⃗ a and ⃗ b is A = | ⃗ a × ⃗ b | : b a the area of the triangle spanned is A = 1 2 | ⃗ a × ⃗ b | : b a Volume of the parallelopiped spanned by ⃗ a , ⃗ b ,⃗ c is...
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## This note was uploaded on 03/26/2012 for the course STAT 350 taught by Professor Staff during the Spring '08 term at Purdue.

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261-S12-sg1 - MA 261 Spring 2012 Study Guide 1 1 Vectors in...

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