s07wk03 - Math 23 B Dodson Week 3 Homework 12.5 Lines Planes 12.6 Quadratic Surfaces 12.7 Cylindrical and Spherical Coords Problem 12.5.3 Give vector

# s07wk03 - 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 12.7 Cylindrical and Spherical Coords 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 + td, 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
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. Problem 12.5.26: Find an equation of the plane through the point ( - 2 , 8 , 10) and perpendicular to the line x = 1 + t, y = 2 t, z = 4 - 3 t.
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