General Notice
I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 12.3 : Equations of Planes
For problems 1 – 3 write down the equation of the plane.
- The plane containing the points \(\left( {4, - 3,1} \right)\), \(\left( { - 3, - 1,1} \right)\) and \(\left( {4, - 2,8} \right)\). Solution
- 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
- 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.
- The plane given by \(4x - 9y - z = 2\) and the plane given by \(x + 2y - 14z = - 6\). Solution
- 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.
- 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
- 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
- Find the line of intersection of the plane given by \(3x + 6y - 5z = - 3\) and the plane given by \( - 2x + 7y - z = 24\). Solution
- 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