Crip Sheet for Final - comp a b=ab a a ab a Vector proj of...

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Scalar proj of b onto a: comp a b = a b / ¿ a ¿ Vector proj of b onto a: ¿ a ¿ proj a b = ( a b | a | ) a ¿ Equations of Lines and Planes: Lines: Directional vector v = B<a,b,c>-A<d,e,f> Planes: n (r - r0) = 0 , n r = n r0 a(x - x0)+b(y - y0)+c(z - z0) = 0 P0 = (x0, y0, z0), normal vector n = <a, b, c> ax + by + cz + d = 0, n = a X b Distance D from a point P1(x1, y1, z1) ¿ a x 1 + b y 1 + c z 1 + d ¿ a 2 + b 2 + c 2 D = ¿ Graphs of Quadric Surfaces: Ellipsoid : x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 Elliptic Paraboloid : x 2 a 2 + y 2 b 2 = z c Hyperbolic Paraboloid : x 2 a 2 y 2 b 2 = z c Cone : x 2 a 2 + y 2 b 2 = z 2 c 2 Hyperboloidof Onesheet : x 2 a 2 + y 2 b 2 z 2 c 2 = 1 Hyperboloidof twosheets : x 2 a 2 y 2 b 2 + z 2 c 2 = 1 Arc Length and Curvature: x = f(t), y = g(t), z = h(t), a<t<b L = a b ( dx dt ) 2 + ( dy dt ) 2 + ( dz dt ) 2 dt Directional Deriv. and the Gradient Vector D u f ( x, y ) = f x ( x, y ) a + f y ( x , y ) b = f ( x, y ) u u = <a, b> << any unit vector Maximum value of the dir. Vec. = ¿ f ( x , y ) ¿ Maximum and Minimum Values (a, b) = critical point if f x ( a,b ) = 0, f y ( a,b ) = 0 .
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