s10wk03 - Math 23 B Dodson Week 3 Homework 12.5 Lines...

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Math 23 B. Dodson Week 3 Homework: 12.5 Lines, Planes 12.6 Quadratic Surfaces 13.1, 13.2 Vector functions, derivatives Problem 12.5.3: Give vector and (scalar) parametric equations for the line through the point (-2,4,10) parallel to the vector < 3 , 1 , 8 > . Solution: The vector equation ± OP = ± OP 0 + t ± d, for when the position vector of the point P ( x, y, z ) puts P on the line through P 0 ( x 0 , y 0 , z 0 ) with direction vector ± d = < a, b, c > gives < x, y, z > = < x 0 , y 0 , z 0 > + t < a, b, c > = < - 2 , 4 , 10 > + t < 3 , 1 , 8 >, which we can view as a “point-slope” equation, where P 0 is the point, and ± d gives the direction of the line. (Here in the position vector ± OP, O = O (0 , 0 , 0) is the Origin.) To get the scalar equations, we use scalar mult. and vector add to write the vector equation
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2 . as < x, y, z > = < - 2 + 3 t, 4 + t, 10 + 8 t >, and simply read-off x = - 2 + 3 t, y = 4 + t, z = 10 + 8 t. Here, each value of the parameter t gives a point on the line.
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s10wk03 - Math 23 B Dodson Week 3 Homework 12.5 Lines...

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