# Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations P(−3, −6, −1), Q(−1, −9, −8).

Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations $P\left(-3,-6,-1\right),Q\left(-1,-9,-8\right)$.
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Two vectors $P\left({x}_{1},{y}_{1},{z}_{1}\right)andQ\left({x}_{2},{y}_{2},{z}_{2}\right)$ Line segment P to Q : 1) Parametric form is $x={x}_{1}+at$
$y={y}_{1}+bt$
$z={z}_{1}+ct$
2) vector valued form is $r=$ Given vectors $P\left(-3,-6,-1\right)\text{and}Q\left(-1,-9,-8\right)$ Line segment P to Q: Firstly evaluate a, b and c $PQ=<{x}_{2}⎯{x}_{1},{y}_{2}⎯{y}_{1},{z}_{2}⎯{z}_{1}>$
$=<\left(⎯1\right)⎯\left(⎯3\right),\left(⎯9\right)⎯\left(⎯6\right),\left(⎯8\right)⎯\left(⎯1\right)>$
$=<⎯1+3,⎯9+6,⎯8+1>$
$=<2,⎯3,⎯7>$ 1) Parametric form is $x={x}_{1}+at$
$=⎯3+\left(2\right)t$
$=⎯3+2t$
$y={y}_{1}+bt$
$=⎯6+\left(⎯3\right)t$
$=⎯6⎯3t$
$z={z}_{1}+ct$
$=⎯1+\left(⎯7\right)t$
$=⎯1⎯7t$ 2) Vector valued function: $r\left(t\right)=<⎯3+2t,⎯6⎯3t,⎯1⎯7t>$
$\left(⎯3+2t\right)i+\left(⎯6⎯3t\right)j+\left(⎯1⎯7t\right)k$ Hence 1) vector valued function is $r\left(t\right)=\left(⎯3+2t\right)i+\left(⎯6⎯3t\right)j+\left(⎯1⎯7t\right)k$ 2) Parametric form is $x=⎯3+2t$
$y=⎯6⎯3t$
$z=⎯1⎯7t$