Step 1

The given function is \(\displaystyle{r}{\left({t}\right)}={3}{t}^{{2}}{i}+{\left({t}-{1}\right)}{j}+{t}{k}\)

Step 2

Here \(\displaystyle{x}{\left({t}\right)}={3}{t}^{{2}},{y}{\left({t}\right)}={t}-{1},{z}{\left({t}\right)}={t}\)

A horizontal tra

ation one unit in the direction of the positive x-axis gives the function x(t) as,

\(\displaystyle{x}{\left({t}\right)}={3}{t}^{{2}}+{1}\)

Thus, \(\displaystyle{r}{\left({t}\right)}={\left({3}{t}^{{2}}+{1}\right)}{i}+{\left({t}-{1}\right)}{j}+{t}{k}\).

The given function is \(\displaystyle{r}{\left({t}\right)}={3}{t}^{{2}}{i}+{\left({t}-{1}\right)}{j}+{t}{k}\)

Step 2

Here \(\displaystyle{x}{\left({t}\right)}={3}{t}^{{2}},{y}{\left({t}\right)}={t}-{1},{z}{\left({t}\right)}={t}\)

A horizontal tra

ation one unit in the direction of the positive x-axis gives the function x(t) as,

\(\displaystyle{x}{\left({t}\right)}={3}{t}^{{2}}+{1}\)

Thus, \(\displaystyle{r}{\left({t}\right)}={\left({3}{t}^{{2}}+{1}\right)}{i}+{\left({t}-{1}\right)}{j}+{t}{k}\).