Step 1

Vector-valued Function represented as

\(\displaystyle\vec{{{r}}}{\left({t}\right)}={X}{\left({t}\right)}\hat{{{i}}}+{Y}{\left({t}\right)}\hat{{{j}}}\)

Step 2

Plane Curve is given as

y=x+1

So, let X(t)=t

then Y(t)=t+1

Therefore, Vector-valued Function is

\(\displaystyle\vec{{{r}}}{\left({t}\right)}={X}{\left({t}\right)}\hat{{{i}}}+{Y}{\left({t}\right)}\hat{{{j}}}\)

\(\displaystyle\vec{{{r}}}{\left({t}\right)}={t}\hat{{{i}}}+{\left({t}+{1}\right)}\hat{{{j}}}\)

Vector-valued Function represented as

\(\displaystyle\vec{{{r}}}{\left({t}\right)}={X}{\left({t}\right)}\hat{{{i}}}+{Y}{\left({t}\right)}\hat{{{j}}}\)

Step 2

Plane Curve is given as

y=x+1

So, let X(t)=t

then Y(t)=t+1

Therefore, Vector-valued Function is

\(\displaystyle\vec{{{r}}}{\left({t}\right)}={X}{\left({t}\right)}\hat{{{i}}}+{Y}{\left({t}\right)}\hat{{{j}}}\)

\(\displaystyle\vec{{{r}}}{\left({t}\right)}={t}\hat{{{i}}}+{\left({t}+{1}\right)}\hat{{{j}}}\)