Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form x=at+b , y=ct+d , z=et+f 2x_1+x_2+x_3=3 -2x_1+x_2-x_3=1

Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form x=at+b , y=ct+d , z=et+f 2x_1+x_2+x_3=3 -2x_1+x_2-x_3=1

Question
Forms of linear equations
asked 2021-02-09
Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form
\(x=at+b , y=ct+d , z=et+f\)
\(2x_1+x_2+x_3=3\)
\(-2x_1+x_2-x_3=1\)

Answers (1)

2021-02-10
The given system of linear equations
\(2x_1+x_2+x_3=3\)
\(-2x_1+x_2+x_3=1\)
We need to determine whether the given system of linear equations represents coincident planes, two parallel planes, or two planes whose intersection is a line.
Let \(n_1 = (2,1,1)\) and \(n_2 = (-2,1,-1)\) be normal vectors of both the equations. Then n, and na are not parallel.
Let \(x_3 = t\), then replace \(x_3 = t\) in the system of linear equations \(2x_1+x_2+t=3 \ \ \ (1)\)
\(-2x_1+x_2+t=1 \ \ \ (2)\)
\((1)+(2) \Rightarrow 2x_2+2t=4 \Rightarrow 2x_2=4-2t \Rightarrow x_2=2-t\) Replace \(x_2 = 2-t\) and \(x_3 =t\) in the firs equation
\(2x_1+2-t+t=3\)
\(2x_1=3-2\)
\(x_1=\frac{1}{2}\)
Hence, the plane intersect in a line and the parametric equations are
\(x_1=\frac{1}{2} , x_2=2-t \text{ and } x_3=t\)
0

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