Calculus III - Vector Functions

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Section 12.6 : Vector Functions

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6. Identify the graph of the vector function without sketching the graph.

\[\vec r\left( t \right) = \left\langle {3\cos \left( {6t} \right), - 4,\sin \left( {6t} \right)} \right\rangle \]

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To 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 2

Now, 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 \(y = - 4\).