Question

The curve \({\rm{r(t) = }}\langle {\rm{2t,3 - t,0}}\rangle \) is a line that passes through the origin.

Short answer

The given statement is false.

Step-by-step solution

The curve will pass through the origin,

The curve

\({\rm{r(t) = }}\langle {\rm{2t,3 - t,0}}\rangle \]

The curve will pass through the origin if and only if can find such that

\({\rm{r(t) = }}\langle {\rm{0,0,0}}\rangle \]

Equating the above two curves,

Equating the above two curves to solving the system of equations

\(\begin{aligned}{*{20}{r}}{{\rm{2t = 0}}}\\{{\rm{3 - t = 0}}}\end{aligned}\)

The first equation implies that \({\rm{t = 0}}\) but the second equation implies that \({\rm{t = 3}}\] therefore the systemhas no solution, so the curve does not pass through the origin. Therefore, the statement is false.