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Section 12.6 : Vector Functions
6. Identify the graph of the vector function without sketching the graph.
\[\vec r\left( t \right) = \left\langle {3\cos \left( {6t} \right), - 4,\sin \left( {6t} \right)} \right\rangle \]Show All Steps Hide All Steps
Start SolutionTo identify the graph of this vector function (without graphing) let’s first write down the set of parametric equations we get from this vector function. They are,
\[\begin{align*}x & = 3\cos \left( {6t} \right)\\ y & = - 4\\ z & = \sin \left( {6t} \right)\end{align*}\] Show Step 2Now, from the \(x\) and \(z\) equations we can see that we have an ellipse in the \(xz\)-plane that is given by the following equation.
\[\frac{{{x^2}}}{9} + {z^2} = 1\]From the \(y\) equation we know that this ellipse will not actually be in the \(xz\)-plane but parallel to the \(xz\)-plane at \(y = - 4\).