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

Phoebe 2020-11-14 Answered
a) Find parametric equations for the line. (Use the parameter t.)
The line of intersection of the planes
x + y + z = 2 and x + z = 0
(x(t), y(t), z(t)) =
b) Find the symmetric equations.
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Expert Answer

Raheem Donnelly
Answered 2020-11-15 Author has 75 answers
Step 1
Given:
The plane equations are x + y + z = 2 and 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=|n1Xn2|=|ijk111101|
=i(10)j(11)+k(01)
=ik
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(ik)
r=ti+2jtk
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
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