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### Section 6-3 : 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.