Q .5. - Calculus

Question

 Let P and Q be distinct points in ${\mathrm{ℝ}}^3$ Provide a stepby-step procedure for finding the equation of the line containing P and Q 

Step-by-Step Solution

Answer

$r\left(t\right)=(a,b,c)+t(\alpha -a,\beta -b,\gamma -c)$

And

$\text{ The line equation as a vector function is }r\left(t\right)=(a+(\alpha -a)t,b+(\beta -b),c+(\gamma -c\left)t\right)\text{. }$

Step 1:Given information

$\text{ Let }P(a,b,c)\text{ and }Q(\alpha ,\beta ,\gamma )\text{ are any two points. }$

Step 2:Explaination

To construct the equation find the direction vector along the line  $\overrightarrow{PQ}$ of the given points.

The direction vector $(d=\alpha -a,\beta -b,\gamma -c)$

To construct the equation choose the point $P$ or $Q$ and construct the equation.

Here take the point$P(a,b,c)$ and the direction vector $d=(\alpha -a,\beta -b,\gamma -c)$

The formula to find the line  $\mathcal{L}$equation as a vector function is as follows,

$r\left(t\right)=P_0+td$ Where,  $P_0$ is the point and $d$ is the direction vector.

For the point$P(a,b,c)$ and the direction vector $\$d=(\alpha -a,\beta -b,\gamma -c)$ the line equation is as follows,

$r\left(t\right)=(a,b,c)+t(\alpha -a,\beta -b,\gamma -c)$

$\text{ The line equation as a vector function is }r\left(t\right)=(a+(\alpha -a)t,b+(\beta -b),c+(\gamma -c\left)t\right)\text{. }$