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261-S11-sg1-1

# 261-S11-sg1-1 - MA 261 Spring 2011 Study Guide 1 1 Vectors...

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MA 261 - Spring 2011 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 0 ( x 0 , y 0 , z 0 ) to P 1 ( x 1 , y 1 , z 1 ) given by v = P 0 P 1 = P 1 P 0 = x 1 x 0 , y 1 y 0 , z 1 z 0 . (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 0 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

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Volume of the parallelopiped spanned by a , b ,⃗ c is V = | a · ( b × c ) | : b a c 2.
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261-S11-sg1-1 - MA 261 Spring 2011 Study Guide 1 1 Vectors...

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