Engineering Calculus Notes 46

Engineering Calculus Notes 46 - 34 CHAPTER 1. COORDINATES...

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Unformatted text preview: 34 CHAPTER 1. COORDINATES AND VECTORS The points lying on the line specified by a particular direction vector v and basepoint P are best described in terms of their position vectors. Denote the position vector of the basepoint by p = x + y + z k ; then to reach any other point P ( x,y,z ) on the line, we travel parallel to v from P , which is to say the displacement P P from P is a scalar multiple of the direction vector: P P = t v . The position vector p ( t ) of the point corresponding to this scalar multiple of v is p ( t ) = p + t v which defines a vector-valued function of the real variable t . In terms of coordinates, this reads x = x + at y = y + bt z = z + ct. We refer to the vector-valued function p ( t ) as a parametrization ; the coordinate equations are parametric equations for the line....
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