13-14-review - 1. A vector function is a function that...

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1. A vector function is a function that assigns to each x a vector r . Just like a regular function assigns a point. 2. A vector function can be represented by a space curve. A space curve is just a plane curve in 3 dimensions. (A vector function can also draw out a plane curve.) 3. A smooth curve (in vector notation) is one whose corresponding vector function’s derivative with respect to t never equals the zero vector. This is done to force the curve’s tangent vector to be continuously turning. You find the tangent vector of a smooth curve at a point by taking the derivative of the general vector equation with respect to t (by taking the derivative of each of its components, then setting these as the components of a new vector), and finally plugging in the point into the component equations. Things to Know : When r t = c (a constant), r is orthogonal to its derivative. Distance formula in R 3 : x h 2 y k 2 z l 2 . Eqn. of a sphere: x h 2 y k 2 z l 2 = r 2 Properties of vector addition/subtraction: x y z x y = z Standard representations: i = 1, 0, 0 j = 0, 1, 0 k = 0, 0, 1 Magnitude of Vector: place vector at origin, then use distance formula: v = x 2 y 2 z 2 Dot product and associated formulas: a b = a 1 b 1 a 2 b 2 a 3 b 3 a a = a 2 a b = b a a b c = a b a c ca b = c a b = a cb 0 a = 0 a b = a b cos = arccos a b a b a b = 0. b
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13-14-review - 1. A vector function is a function that...

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