Paul's Online Notes
Paul's Online Notes
Home / Calculus III / Surface Integrals / Parametric Surfaces
Show Mobile Notice Show All Notes Hide All Notes
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 views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 17.2 : Parametric Surfaces

1. Write down a set of parametric equations for the plane \(7x + 3y + 4z = 15\).

Show All Steps Hide All Steps

Start Solution

There isn’t a whole lot to this problem. There are three different acceptable answers here. To get a set of parametric equations for this plane all we need to do is solve for one of the variables and then write down the parametric equations.

For this problem let’s solve for \(z\) to get,

\[z = \frac{{15}}{4} - \frac{7}{4}x - \frac{3}{4}y\] Show Step 2

The parametric equation for the plane is then,

\[\vec r\left( {x,y} \right) = \left\langle {x,y,z} \right\rangle = \require{bbox} \bbox[2pt,border:1px solid black]{{\left\langle {x,y,\frac{{15}}{4} - \frac{7}{4}x - \frac{3}{4}y} \right\rangle }}\]

Remember that all we need to do to get the parametric equations is plug in the equation for \(z\) into the \(z\) component of the vector \(\left\langle {x,y,z} \right\rangle \).

Also, as noted in Step 1 we could just have easily done either of the following two forms for the parametric equations for this plane.

\[\vec r\left( {x,z} \right) = \left\langle {x,g\left( {x,z} \right),z} \right\rangle \hspace{0.5in}\vec r\left( {y,z} \right) = \left\langle {h\left( {y,z} \right),y,z} \right\rangle \]

where you solve the equation of the plane for \(y\) or \(x\) respectively. All three set of parametric equations are all perfectly valid forms for the answer to this problem.