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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