Find parametric equations and symmetric equations for the line. The line of intersection of the planes x+2y+3z=1 and x−y+z=1

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Find parametric equations and symmetric equations for the line.  The line of intersection of the planes  x+2y+3z=1  and  x−y+z=1

Uersfeldte · Accepted Answer

Step 1&amp;nbsp;Formula:&amp;nbsp;Parametric equations are given by&amp;nbsp;x=x0+a&amp;nbsp;task&amp;nbsp;y=y0+b&amp;nbsp;task&amp;nbsp;z=z0+c&amp;nbsp;taskSymmetric equations are given by&amp;nbsp;x−x0a=y−y0b=z−z0c&amp;nbsp;Explanation:&amp;nbsp;To find the parametric and symmetric equations the given equations then,&amp;nbsp;Consider a point&amp;nbsp;&amp;lt;x,&amp;nbsp;y,&amp;nbsp;z&amp;gt;&amp;nbsp;must satisfies the both equations&amp;nbsp;x+2y+3y=1&amp;nbsp;and&amp;nbsp;x−y+z=1&amp;nbsp;1)&amp;nbsp;x+2y+3y=1&amp;nbsp;2)&amp;nbsp;x−y+z=1&amp;nbsp;Step 2&amp;nbsp;For finding the values of&amp;nbsp;x0,&amp;nbsp;y0,&amp;nbsp;z0&amp;nbsp;To find the intersection point:&amp;nbsp;Let&amp;nbsp;x=0&amp;nbsp;or&amp;nbsp;y=0&amp;nbsp;or&amp;nbsp;z=0&amp;nbsp;Consider here&amp;nbsp;z=0&amp;nbsp;then&amp;nbsp;Equation (1) and (2) implies,&amp;nbsp;3)&amp;nbsp;x+2y=1&amp;nbsp;4)&amp;nbsp;x−y=1&amp;nbsp;Now solve these two simultaneous equations&amp;nbsp;Multiply equation (4) by 2 and add it in equation (3)&amp;nbsp;Therefore (4) implies,&amp;nbsp;2x−2y=2&amp;nbsp;x+2y=1&amp;nbsp;+2x−2y=2―&amp;nbsp;3x=3&amp;nbsp;x=1&amp;nbsp;Now put&amp;nbsp;x=1&amp;nbsp;in Equation (3)&amp;nbsp;1+2y=1&amp;nbsp;2y=1−1&amp;nbsp;2y=0&amp;nbsp;y=0&amp;nbsp;Therefore the point of intersection is&amp;nbsp;&amp;lt;1,&amp;nbsp;0,&amp;nbsp;0≥&amp;lt;x0,&amp;nbsp;y0,&amp;nbsp;z0&amp;gt;&amp;nbsp;respectively.&amp;nbsp;Step 3&amp;nbsp;For finding the values of a, b, c :&amp;nbsp;Find the vector that is perpendicular to the planes&amp;nbsp;1)&amp;nbsp;x+2y+3y=1&amp;nbsp;2)&amp;nbsp;x−y+z=1&amp;nbsp;P^1=⟨1,&amp;nbsp;2,&amp;nbsp;3⟩&amp;nbsp;and&amp;nbsp;P^2=⟨1,&amp;nbsp;−1,&amp;nbsp;1⟩&amp;nbsp;P^1×P^2=[i^j^k^1231−11]&amp;nbsp;=i^[23−11]−j^[1311]+k^[121−1]&amp;nbsp;=i^[2(1)−3(−1)]−j^[1(1)−3(1)]+k^[1(−1)−2(1)]&amp;nbsp;&amp;nbsp;

Find parametric equations and symmetric equations for the line. The line

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