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Expert-verified Found in: Page 830 ### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280 # 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.

The given statement is true.

See the step by step solution

## Step 1: Reparametrize the curve,

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

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

## Step 2: 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. ### Want to see more solutions like these? 