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StudyGuide1 - MA 261 Fall 2009 Study Guide 1 1 Vectors in...

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MA 261 - Fall 2009 Study Guide # 1 1. Vectors in R 2 and R 3 (a) ~ v = h a, b, c i = a ~ i + b ~ j + c ~ k ; vector addition and subtraction geometrically using paral- lelograms spanned by ~ u and ~ v ; length or magnitude of ~ v = h a, b, c i , | ~ 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 = h x 1 - x 0 , y 1 - y 0 , z 1 - z 0 i . (b) Dot (or inner) product of ~ a = h a 1 , a 2 , a 3 i and ~ b = h b 1 , b 2 , b 3 i : ~ 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 = h r cos θ, r sin θ i : 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. Equation of a line L through P 0 ( x 0 , y 0 , z 0 ) with direction vector ~ d = h a, b, c i : Vector Form : ~ r ( t ) = h x 0 , y 0 , z 0 i + t ~ d . (x ,y ,z ) 0 0 0 d Parametric Form : x = x 0 + a t y = y 0 + b t z = z 0 + c t Symmetric Form : x - x 0 a = y - y 0 b = z - z 0 c . (If say b = 0, then x - x 0 a = z - z 0 c , y = y 0 . ) 3. Equation of the plane through the point P 0 ( x 0 , y 0 , z 0 ) and perpendicular to the vector ~ n = h a, b, c i ( ~ n is a normal vector to the plane) is h ( x - x 0 ) , ( y - y 0 ) , ( z - z 0 ) i · ~ n = 0; Sketching planes (consider x, y, z intercepts).
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