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Section 6-3 : Equations of Planes

For problems 1 – 3 write down the equation of the plane.

  1. The plane containing the points \(\left( {4, - 3,1} \right)\), \(\left( { - 3, - 1,1} \right)\) and \(\left( {4, - 2,8} \right)\). Solution
  2. The plane containing the point \(\left( {3,0, - 4} \right)\) and orthogonal to the line given by \(\vec r\left( t \right) = \left\langle {12 - t,1 + 8t,4 + 6t} \right\rangle \). Solution
  3. The plane containing the point \(\left( { - 8,3,7} \right)\) and parallel to the plane given by \(4x + 8y - 2z = 45\). Solution

For problems 4 & 5 determine if the two planes are parallel, orthogonal or neither.

  1. The plane given by \(4x - 9y - z = 2\) and the plane given by \(x + 2y - 14z = - 6\). Solution
  2. The plane given by \( - 3x + 2y + 7z = 9\) and the plane containing the points \(\left( { - 2,6,1} \right)\), \(\left( { - 2,5,0} \right)\) and \(\left( { - 1,4, - 3} \right)\). Solution

For problems 6 & 7 determine where the line intersects the plane or show that it does not intersect the plane.

  1. The line given by \(\vec r\left( t \right) = \left\langle { - 2t,2 + 7t, - 1 - 4t} \right\rangle \) and the plane given by \(4x + 9y - 2z = - 8\). Solution
  2. The line given by \(\vec r\left( t \right) = \left\langle {4 + t, - 1 + 8t,3 + 2t} \right\rangle \) and the plane given by \(2x - y + 3z = 15\). Solution
  3. Find the line of intersection of the plane given by \(3x + 6y - 5z = - 3\) and the plane given by \( - 2x + 7y - z = 24\). Solution
  4. Determine if the line given by \(x = 8 - 15t\), \(y = 9t\), \(z = 5 + 18t\) and the plane given by \(10x - 6y - 12z = 7\) are parallel, orthogonal or neither. Solution