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Section 12.3 : Equations of Planes

For problems 1 – 5 write down the equation of the plane.

1. The plane containing the points \(\left( {6, - 3,1} \right)\), \(\left( {5, - 4,1} \right)\) and \(\left( {3, - 4,0} \right)\).
2. The plane containing the point \(\left( {1, - 5,8} \right)\) and orthogonal to the line given by \(x = - 3 + 15t\), \(y = 14 - t\), \(z = 9 - 3t\).
3. The plane containing the point \(\left( { - 8,3,7} \right)\) and parallel to the plane given by \(4x + 8y - 2z = 45\).
4. The plane containing the point \(\left( {2,0, - 8} \right)\) and containing the line given by \(\vec r\left( t \right) = \left\langle {8t, - 1 - 5t,4 - t} \right\rangle \).
5. 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.

1. The plane given by \( - 5x + 3y + 2z = - 8\) and the plane given by \(6x - 8z = 15\).
2. 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 \(\left( { - 2,3,4} \right)\).
3. 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.

1. The line given by \(\vec r\left( t \right) = \left\langle {9 + t, - 4 + t,2 + 5t} \right\rangle \) and the plane given by \(4x - 9y + z = 6\).
2. The line given by \(\vec r\left( t \right) = \left\langle {2 - 3t,1 + t, - 4 - 2t} \right\rangle \) and the plane given by \(x - 7y - 4z = - 1\).
3. 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.

1. Find the line of intersection of the plane given by \(4x + y + 10z = - 2\) and the plane given by \( - 8x + 2y + 3z = - 8\).
2. Find the line of intersection of the plane given by \(x - 10y - 2z = 3\) and the plane given by \(2x - y + z = - 13\).
3. 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.