emancipezN

2021-02-25

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

un4t5o4v

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 $x=⎯3+2t$
$y=⎯6⎯3t$
$z=⎯1⎯7t$

nick1337

To represent the line segment from P to Q using a vector-valued function and a set of parametric equations, we can use the following formulations:
1. Vector-Valued Function:
$𝐫\left(t\right)=𝐏+t\left(𝐐-𝐏\right)$
2. Parametric Equations:
$x={x}_{P}+t\left({x}_{Q}-{x}_{P}\right)$,
$y={y}_{P}+t\left({y}_{Q}-{y}_{P}\right)$,
$z={z}_{P}+t\left({z}_{Q}-{z}_{P}\right)$,
where $𝐏\left(-3,-6,-1\right)$ and $𝐐\left(-1,-9,-8\right)$ are the given points, and $t$ is a parameter representing the position along the line segment.

Vasquez

$\begin{array}{cc}\hfill x\left(t\right)& =-3+2t\hfill \\ \hfill y\left(t\right)& =-6-3t\hfill \\ \hfill z\left(t\right)& =-1-7t\hfill \end{array}$
Explanation:
Let's start with the vector-valued function. We can define a position vector $𝐫\left(t\right)$ that represents any point on the line segment. The vector $𝐫\left(t\right)$ will have components $x\left(t\right)$, $y\left(t\right)$, and $z\left(t\right)$, which are functions of the parameter $t$.
Using the two given points, we can find the direction vector of the line segment by subtracting the coordinates of $P$ from the coordinates of $Q$. The direction vector will give us the change in position as we move along the line segment.
Let's denote the direction vector as $𝐯$:
$𝐯=\left(\begin{array}{c}-1-\left(-3\right)\\ -9-\left(-6\right)\\ -8-\left(-1\right)\end{array}\right)=\left(\begin{array}{c}2\\ -3\\ -7\end{array}\right)$
Now, we can write the vector-valued function as:
$𝐫\left(t\right)=\left(\begin{array}{c}-3\\ -6\\ -1\end{array}\right)+t\left(\begin{array}{c}2\\ -3\\ -7\end{array}\right)$
Next, let's find the set of parametric equations for the line segment. We can express the coordinates $x$, $y$, and $z$ in terms of the parameter $t$:
$x\left(t\right)=-3+2t$
$y\left(t\right)=-6-3t$
$z\left(t\right)=-1-7t$
Thus, the set of parametric equations for the line segment is:
$\begin{array}{cc}\hfill x\left(t\right)& =-3+2t\hfill \\ \hfill y\left(t\right)& =-6-3t\hfill \\ \hfill z\left(t\right)& =-1-7t\hfill \end{array}$

RizerMix

Let's denote the vector-valued function as $𝐫\left(t\right)$ and the parametric equations as $x\left(t\right)$, $y\left(t\right)$, and $z\left(t\right)$.
The direction vector, $𝐝$, is given by:
$𝐝=𝐐-𝐏$
Substituting the given coordinates, we have:
$𝐝=\left(\begin{array}{c}-1\\ -9\\ -8\end{array}\right)-\left(\begin{array}{c}-3\\ -6\\ -1\end{array}\right)=\left(\begin{array}{c}-1-\left(-3\right)\\ -9-\left(-6\right)\\ -8-\left(-1\right)\end{array}\right)=\left(\begin{array}{c}2\\ -3\\ -7\end{array}\right)$
Now, we can represent the line segment using the vector-valued function:
$𝐫\left(t\right)=𝐏+t𝐝$
Substituting the coordinates and the direction vector, we get:
$𝐫\left(t\right)=\left(\begin{array}{c}-3\\ -6\\ -1\end{array}\right)+t\left(\begin{array}{c}2\\ -3\\ -7\end{array}\right)=\left(\begin{array}{c}-3+2t\\ -6-3t\\ -1-7t\end{array}\right)$
The line segment can also be represented by the set of parametric equations:
$\begin{array}{c}\hfill x\left(t\right)=-3+2t\\ \hfill y\left(t\right)=-6-3t\\ \hfill z\left(t\right)=-1-7t\end{array}$
These equations describe the coordinates of points on the line segment as a function of the parameter t. By varying the value of t within a suitable range, we can traverse the line segment from point P to point Q.

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