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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 12.3 : Equations of Planes
For problems 1 – 5 write down the equation of the plane.
- The plane containing the points \(\left( {6, - 3,1} \right)\), \(\left( {5, - 4,1} \right)\) and \(\left( {3, - 4,0} \right)\).
- The plane containing the point \(\left( {1, - 5,8} \right)\) and orthogonal to the line given by \(x = - 3 + 15t\), \(y = 14 - t\), \(z = 9 - 3t\).
- The plane containing the point \(\left( { - 8,3,7} \right)\) and parallel to the plane given by \(4x + 8y - 2z = 45\).
- The plane containing the point \(\left( {2,0, - 8} \right)\) and containing the line given by \(\vec r\left( t \right) = \left\langle {8t, - 1 - 5t,4 - t} \right\rangle \).
- The plane containing the two lines given by \(\vec r\left( t \right) = \left\langle {7 + 5t,2 + t,6t} \right\rangle \) and \(\vec r\left( t \right) = \left\langle {7 - 6t,2 - 2t,10t} \right\rangle \).
For problems 6 – 8 determine if the two planes are parallel, orthogonal or neither.
- The plane given by \( - 5x + 3y + 2z = - 8\) and the plane given by \(6x - 8z = 15\).
- The plane given by \(3x + 9y + 7z = - 1\) and the plane containing the points \(\left( {1, - 1,9} \right)\), \(\left( {4, - 1,2} \right)\) and \(\left( { - 2,3,4} \right)\).
- The plane given by \( - x - 8y + 3z = 6\) and the plane given by \(2x + 2y + 6z = - 91\).
For problems 9 – 11 determine where the line intersects the plane or show that it does not intersect the plane.
- The line given by \(\vec r\left( t \right) = \left\langle {9 + t, - 4 + t,2 + 5t} \right\rangle \) and the plane given by \(4x - 9y + z = 6\).
- The line given by \(\vec r\left( t \right) = \left\langle {2 - 3t,1 + t, - 4 - 2t} \right\rangle \) and the plane given by \(x - 7y - 4z = - 1\).
- The line given by \(x = 8\), \(y = - 9t\), \(z = 1 + 10t\) and the plane given by \(8x + 9y + 2z = 17\).
For problems 12 & 13 find the line of intersection of the two planes.
- Find the line of intersection of the plane given by \(4x + y + 10z = - 2\) and the plane given by \( - 8x + 2y + 3z = - 8\).
- Find the line of intersection of the plane given by \(x - 10y - 2z = 3\) and the plane given by \(2x - y + z = - 13\).
- Determine if the line given by \(x = 4 + 3t\), \(y = - 2\), \(z = 1 + 6t\) and the plane given by \(8x - y + 4z = - 3\) are parallel, orthogonal or neither.