261-S12-sg1

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

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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 = a b | a || b | ; a b if and only if a b = 0; the vector in R 2 with length r with angle is v = r cos , r sin : x y 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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261-S12-sg1 - MA 261 - Spring 2012 Study Guide # 1 1....

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