Section 12.3 : 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