Q 19. - Calculus

Question

Let L be the line determined by the system of equations

$x\left(t\right) = 4, y\left(t\right) = 3 - 5t, z\left(t\right) = t,-\infty < t < \infty$

(a) Provide a vector parametrization for $L$

(b) Write an equation for $L$ in symmetric form.

Step-by-Step Solution

Answer

Part (a) $r\left(t\right)=(4,3-5 t, t)$

Part (b) $x=4,\frac{y-3}{-5}=z$

Part (a) Step 1: Given information

Consider the line $\mathcal{L}$ determined by the equations  $x\left(t\right)=4, y\left(t\right)=3-5 t, z\left(t\right)=t$ $-\infty \le t\le \infty$

Part (a) Step 2: Explanation

The goal is to use vector parametrization to express the equation $\mathcal{L}$

Given equations of the line $\mathcal{L}$ are $x\left(t\right)=4, y\left(t\right)=3-5 t, z\left(t\right)=t$

The vector parametrization of the line $\mathcal{L}$ is represented by $r\left(t\right)=\left(x\right(t), y(t), z(t\left)\right)$

Then, $r\left(t\right)=\left(x\right(t), y(t), z(t\left)\right)=(4,3-5 t, t)$

Thus, the vector parameterization is $r\left(t\right)=(4,3-5 t, t)$

Therefore, the answer is $r\left(t\right)=(4,3-5 t, t)$

Part (b) Step 1: Explanation

The goal is to write the symmetric form of the equation $\mathcal{L}$

Remove the parameter $t$ from the parametric equations of the line $L$ to write the symmetric form.

The parametric equations are $x\left(t\right)=4, y\left(t\right)=3-5 t, z\left(t\right)=t$

Take $x\left(t\right)=4$

$x=4\dots \dots \left(1\right)$

Take $y\left(t\right)=3-5 t$

$y=3-5 t$

On both sides of the equation, add $-3$

$$\begin{gathered} y-3=3-5t-3 \\ y-3=\backslash \mathrm{not}\beta -5t-\backslash \mathrm{not}\beta \\ y-3=-5t \end{gathered}$$

On both sides of the equation, multiply by $-5$

$$\begin{gathered} \frac{y-3}{-5}=\frac{-5t}{-5} \\ \frac{y-3}{-5}=t\dots \dots \left(2\right) \end{gathered}$$

Take $z\left(t\right)=t$

$z=t\text{......(3) }$

By equating the equations $\left(2\right),\left(3\right)$ that is $\frac{y-3}{-5}=t,z=t$ and $\left(1\right) x=4$ they can be written in the following way.

$x=4,\frac{y-3}{-5}=z=t$

Thus, the symmetric equations are $x=4,\frac{y-3}{-5}=z$

Therefore, the required answer is $x=4,\frac{y-3}{-5}=z$