Question

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases.

\({\bf{r}}(t) = \left\langle {{t^3},{t^2}} \right\rangle \)

Short answer

The curve of vector equation \({\bf{r}}(t) = \left\langle {{t^3},{t^2}} \right\rangle \) indicates that the left side of the curve decreases towards origin while right side of the curve increases from origin.

Step-by-step solution

Parametric equations to plot space curve

Consider a vector function as\({\bf{r}}(t) = \langle f(t),g(t)\rangle \), then parametric equations to plot space curve\({\bf{C}}\)are\(x = f(t)\)and\(y = g(t)\).

To sketch the curve of vector equation

By the definition the parametric equations for vector function \(r(t) = \left\langle {{t^3},{t^2}} \right\rangle \) are \(x = {t^3}\) and \(y = {t^2}\).

Consider a certain interval of \(t\) as \(( - 4,4)\) to plot the vector equation.

The estimated values of \(x\) and \(y\) in the interval \(( - 4,4)\) are shown in Table 1.

From the table, the sketch of the curve of vector equation along with increasing direction of t is drawn as shown in the figure.

Thus, the curve of vector equation \({\bf{r}}(t) = \left\langle {{t^3},{t^2}} \right\rangle \) indicates that the left side of the curve decreases towards origin while right side of the curve increases from origin.