Find parametric equations for the line. The line of intersection of the planes x+y+z=2 and x+z=0 (x(t), y(t), z(t))=? Find the symmetric equations.

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Find parametric equations for the line.  The line of intersection of the planes  x+y+z=2 and x+z=0  (x(t), y(t), z(t))=?  Find the symmetric equations.

Linda Tincher · Accepted Answer

Step 1 Given: The plane equations are x+y+z=2 x+z=0 To find: (a) The parametric equations for the line of intersection of the planes x+y+z=2 and x+z=0 (b)The symmetric equations. Step 2 a) Consider the planes x+y+z=2 and x+z=0 To find the vector equation of the line of intersection, we need to find the cross product v of the normal vectors of the given planes and a point on the line of intersection. The normal vector for the plane x+y+z=2 is n1→=(111) The normal vector for the plane x+z=0 is n2→=(101) Step 3 The cross product of the normal vectors is v→=|n1→Xn2→|=|i→j→k→111101| =i→(1−0)−j→(1−1)+k→(0−1) i→−k→ To find a point on the line of intersection, put z=0 in both the plane equations, x+y+z=2 and x+z=0 we get x+y=2 and x=0 That is x=0 and y=2 Therefore, the point of intersection is r0=(0, 2, 0) That is r0=0i→+2j→+0k→=2j→ Step 4 The vector equation is given by, r=r0+tv r=2j→+t(i→−k→) r=ti→+2j→−tk→ Therefore, the parametric equations for the line of intersection of the planes are x=t, y=2, z=−t Step 5 (b) The symmetric equations: To find the symmetric equation, we solve each of the parametric equations for t and then set them equal. t=x, y=2, t=−z Setting them equal gives us the symmetric form: x=−z and y=2

Find parametric equations for the line. The line of intersection of

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