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topics_13_5_13_6_1 - 13.5. Equations of lines and planes...

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13.5. Equations of lines and planes Vector equation of a line. Consider a line L with a point P 0 ( x 0 ,y 0 ,z 0 ) from the line and a vector v parallel to the line. Let P ( x,y,z ) be an arbitrary point from the line L . Denote by r 0 the position vector corre- sponding to P 0 and by r the position vector corresponding to P . Then, there is a scalar t such that r = r 0 + t v , -∞ < t < , where t is a parameter and equivalently in terms of components of vectors: h x , y , z i = h x 0 , y 0 , z 0 i + t h v 1 , v 2 , v 3 i . The above equation is called a vector equation of the line L . v is called a directional vector of L and its components v = h v 1 ,v 2 ,v 3 i are called direction numbers of L . Parametric equation of a line. If we equal the corresponding com- ponents in the above vector equation we obtain parametric equations of a line: x = x 0 + v 1 t y = y 0 + v 2 t z = z 0 + v 3 t. This is one-parameter ( t is the parameter) representation of the line L passing through the points P 0 ( x 0 ,y 0 ,z 0 ) and parallel to the directional vector v . The vector equation and the parametric equations of a line are not unique. They depend on the choice of P 0 and v. Exercise. Suppose that the line L is on the xy -plane. Then what will be the form of the above parametric equations? Symmetric equation of a line. Eliminating the parameter t we obtain symmetric equations of the line L : x - x 0 v 1 = y - y 0 v 2 = z - z 0 v 3 1
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Problem 6/page 838. Find a vector equation, parametric equations, and symmetric equations for the line passing through the origin and the point (1 , 2 , 3). Problem 10/page 838. Find parametric equations and symmetric equations for the line passing through the point (2 , 1 , 0) and perpen- dicular to both i + j and j + k . Problem 12/page 838. Find parametric equations and symmetric equations for the line of intersection of the planes x + y + z = 1 and x + z = 0. Problem 16/page 838. (a) Find parametric equations, and sym- metric equations for the line passing through the point (2 , 4 , 6) that is perpendicular to the plane x - y + 3 z = 7. (b) In what points does the line intersects the coordinate planes? Problems 19,20/page 838.
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This note was uploaded on 02/05/2010 for the course MATH 264 taught by Professor Dr.d.dryanov during the Fall '09 term at Concordia Canada.

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topics_13_5_13_6_1 - 13.5. Equations of lines and planes...

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