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Study guide 1

# Study guide 1 - MA 261 Fall 2010 Study Guide 1 1 Vectors in...

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Unformatted text preview: MA 261 - Fall 2010 Study Guide # 1 1. Vectors in R 2 and R 3 (a) vector v = ( a, b, c ) = a vector i + b vector j + c vector k ; vector addition and subtraction geometrically using paral- lelograms spanned by vector u and vector v ; length or magnitude of vector v = ( a, b, c ) , | vector 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 vector v = P P 1 = P 1 − P = ( x 1 − x , y 1 − y , z 1 − z ) . (b) Dot (or inner) product of vector a = ( a 1 , a 2 , a 3 ) and vector b = ( b 1 , b 2 , b 3 ) : vector a · vector b = a 1 b 1 + a 2 b 2 + a 3 b 3 ; properties of dot product; useful identity: vector a · vector a = | vector a | 2 ; angle between two vectors vector a and vector b : cos θ = vector a · vector b | vector a || vector b | ; vector a ⊥ vector b if and only if vector a · vector b = 0; the vector in R 2 with length r with angle θ is vector v = ( r cos θ, r sin θ ) : x y θ r (c) Projection of vector b along vector a : proj vector a vector b = braceleftBigg vector a · vector b | vector a | bracerightBigg vector a | vector a | ; Work= vector F · vector D . b proj a proj a b b a a b (d) Cross product (only for vectors in R 3 ): vector a × vector b = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vector i vector j vector k a 1 a 2 a 3 b 1 b 2 b 3 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = vextendsingle vextendsingle vextendsingle vextendsingle a 2 a 3 b 2 b 3 vextendsingle vextendsingle vextendsingle vextendsingle vector i − vextendsingle vextendsingle vextendsingle vextendsingle a 1 a 3 b 1 b 3 vextendsingle vextendsingle vextendsingle vextendsingle vector j + vextendsingle vextendsingle vextendsingle vextendsingle a 1 a 2 b 1 b 2 vextendsingle vextendsingle vextendsingle vextendsingle vector k properties of cross products; vector a × vector b is perpendicular...
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Study guide 1 - MA 261 Fall 2010 Study Guide 1 1 Vectors in...

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