Q 21. - Calculus

Question

find the equation of the line containing the given points in slope-intercept form. Then, use the technique of this section to find a vector parametrization for the same line. Finally, show that your equations are equivalent.

$P(0, 5), Q(2,-1)$

Step-by-Step Solution

Answer

Slope intercept form $y=-3 x+5 ; x\left(t\right)=2 t, y\left(t\right)=-3 x+5$ equivalent.

Step 1: Given information

$P(0,5), Q(2,-1)$

Step 2: Calculation

The goal is to determine the line equation in both slope-intercept and vector parametrization form and prove that they are the same.

The slope-intercept form of the equation is, $y=m x+b$ where $m$ is the slope and $b$ is the $y$ intercept.

Find the slope of the given points first.

According to the formula, $m=\frac{y_2-y_1}{x_2-x_1}$

For the points, $P(0,5), Q(2,-1)$ the slope is as follows,

$$\begin{gathered} m=\frac{-1-5}{2-0}\left[\text{ since }{x}_1=0,y_1=5,x_2=2,y_2=-1\right] \\ m=\frac{-6}{2} \end{gathered}$$

$m=-3$

Now substitute the slope $m=-3$ and the point $P(0,5)$ in $y=m x+b$

$$\begin{gathered} 5=-3·0+b \\ 5=b \\ b=5 \end{gathered}$$

By substitution of $m=-3, b=5$ in the equation $y=m x+b$

$y=-3·x+5$

Thus the line equation is $y=-3 x+5$

Now find the line equation using the vector function.

The points are $P(0,5), Q(2,-1)$

We'll start by determining the direction vector of the line $\overrightarrow{PQ}$

The points are $P(0,5)$ and $Q(2,-1)$

$$\begin{gathered} \overrightarrow{PQ}=(2-0,-1-5) \\ \overrightarrow{PQ}=(2,-6) \end{gathered}$$