Vector equation of a line segment joining the points with position vectors \(r_0\) and \(r_1\) is

\(r=(1-t)r_0+tr_1\)

Where \(t\in[0,1]\)

Substitute \(r_0=<0,-1,1>\) and \(r_1=<\frac{1}{2}.\frac{1}{3},\frac{1}{4}>\)

\(r(t)=(1-t)(0,-1,1)+t<\frac{1}{2},\frac{1}{3},\frac{1}{4}>\)

\(r(t)=<0,-1+t,1-t>+<\frac{t}{2},\frac{t}{3},\frac{t}{4}>\)

\(r(t)=<\frac{t}{2},-1+\frac{4t}{3},1-\frac{3t}{4}>\)

Where \(t\in[0,1]\) The parametric equations for the line segment are

\(x=\frac{t}{2},y=-1+\frac{4t}{3},z=1-\frac{3t}{4}\)

Where \(t\in[0,1]\)

\(r=(1-t)r_0+tr_1\)

Where \(t\in[0,1]\)

Substitute \(r_0=<0,-1,1>\) and \(r_1=<\frac{1}{2}.\frac{1}{3},\frac{1}{4}>\)

\(r(t)=(1-t)(0,-1,1)+t<\frac{1}{2},\frac{1}{3},\frac{1}{4}>\)

\(r(t)=<0,-1+t,1-t>+<\frac{t}{2},\frac{t}{3},\frac{t}{4}>\)

\(r(t)=<\frac{t}{2},-1+\frac{4t}{3},1-\frac{3t}{4}>\)

Where \(t\in[0,1]\) The parametric equations for the line segment are

\(x=\frac{t}{2},y=-1+\frac{4t}{3},z=1-\frac{3t}{4}\)

Where \(t\in[0,1]\)