Question

The curve with the vector equation \({\rm{r(t) = }}{{\rm{t}}^{\rm{3}}}{\rm{i + 2}}{{\rm{t}}^{\rm{3}}}{\rm{j + 3}}{{\rm{t}}^{\rm{3}}}{\rm{k}}\) is a line.

Short answer

The given statement is true.

Step-by-step solution

Reparametrize the curve,

The parameter always appears with its third power. If redistributing the curve in this way, they can get a better result

\({\rm{r(t) = }}{{\rm{t}}^{\rm{3}}}{\rm{i + 2}}{{\rm{t}}^{\rm{3}}}{\rm{j + 3}}{{\rm{t}}^{\rm{3}}}{\rm{k}}\)

Put \({{\rm{t}}^{\rm{3}}}{\rm{ = s}}\) in the given equation,          

Let, \({{\rm{t}}^{\rm{3}}}{\rm{ = s}}\)

\({\rm{r(s) = si + 2sj + 3sk}}\)

This can also write as

\({\rm{r(s) = s}}\langle {\rm{1,2,3}}\rangle \)

The curve is a line whose direction is specified by the vector, as we can see from this form \(\langle {\rm{1,2,3}}\rangle \)

Therefore, the statement is true.