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Section 6-2 : Parametric Surfaces

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

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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.