Question

Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q\).

\(P\left( { - 1,2, - 2} \right)\,\;\;Q\left( { - 3,5,1} \right)\)

Short answer

The vector equation and parametric equations of line segment joining \(P\) to \(Q\) are \(r(t) = \left\langle { - 1 - 2t,2 + 3t, - 2 + 3t} \right\rangle \) and \(x = - 1 - 2t,y = 2 + 3t\), and \(z = - 2 + 3t\) respectively.

Step-by-step solution

Line Segment from Two Vector

Write the expression for line segment from two vectors \({r_0}\) to \({r_1}(r(t))\).

\(r(t) = (1 - t){r_0} + t{r_1}\;\;\; \ldots \left( 1 \right)\)for \(0 \le t \le 1\)

Here, \(t\) is parameter.

Consider a vector equation as \(r(t) = \left\langle {f(t),g(t),h(t)} \right\rangle \), then parametric equations to plot space curve \(C\) are,

\(x = f\left( t \right)\)

\(y = g(t)\)

\(z = h\left( t \right)\)\(\)

Here, \(f\left( t \right),g\left( t \right)\), and \(h\left( t \right)\) are component functions of \(r\left( t \right)\), and , \(x,y\) and \(z\) are parametric equations of \(C\).

Find the line segment \(r\left( t \right)\) joining \(P\) to \(Q\) by the use of equation (1).

\(\begin{aligned}{c}r(t) &= (1 - t)\left\langle { - 1,2, - 2} \right\rangle + t\left\langle { - 3,5,1} \right\rangle \\ &= \left\langle { - 1 + t,2 - 2t, - 2 + 2t} \right\rangle + \left\langle { - 3t,5t,t} \right\rangle \\ &= \left\langle { - 1 + t - 3t,2 - 2t + 5t, - 2 + 2t + t} \right\rangle \\ &= \left\langle { - 1 - 2t,2 + 3t, - 2 + 3t} \right\rangle \end{aligned}\)

Find parametric equation

From formula, write the parametric equations of the vector equation:

\(r(t) = \left\langle { - 1 - 2t,2 + 3t, - 2 + 3t} \right\rangle \)

\(\begin{aligned}{c}x &= f(t)\\ &= - 1 - 2t\end{aligned}\)

\(\begin{aligned}{c}y &= g(t)\\ &= 2 + 3t\end{aligned}\)

\(\begin{aligned}{c}z &= h(t)\\ &= - 2 + 3t\end{aligned}\)

Thus, The vector equation and parametric equations of line segment joining \(P\) to \(Q\) are \(r(t) = \left\langle { - 1 - 2t,2 + 3t, - 2 + 3t} \right\rangle \) and \(x = - 1 - 2t,y = 2 + 3t\), and \(z = - 2 + 3t\) respectively.