Question

Find the vector and Cartesian equation of a line joining the points B(4,7,1) and C(3,5,3).​

Answer 1

Given:

A line joining the points B(4,7,1) & C(3,5,3)

To Find:

The vector & cartesian equation

Solution:

let $\overrightarrow{a}$ & $\overrightarrow{b}$ be the position vectors of the points B(4,7,1) & C (3,5,3).

     $\overrightarrow{a}$= 4$\hat{i}$+7$\hat{j}$+$\hat{k}$   ,$\overrightarrow{b}$= 3$\hat{i}$ +5$\hat{j}$+3$\hat{k}$

VECTOR EQUATION:

The vector equation of the line is,

$\overrightarrow{r}$=$\overrightarrow{a}$+λ($\overrightarrow{b}$-$\overrightarrow{a}$)      ---------(1)

First we find the value of ($\overrightarrow{b}$-$\overrightarrow{a}$),

($\overrightarrow{b}$-$\overrightarrow{a}$)=(3$\hat{i}$ +5$\hat{j}$+3$\hat{k}$) - (4$\hat{i}$+7$\hat{j}$+$\hat{k}$)

($\overrightarrow{b}$-$\overrightarrow{a}$)=-$\hat{i}$-2$\hat{j}$+3$\hat{k}$

By substituting the value of  $\overrightarrow{a}$  & ($\overrightarrow{b}$-$\overrightarrow{a}$) in equation (1),we get

$\overrightarrow{r}$=4$\hat{i}$+7$\hat{j}$+$\hat{k}$+λ(-$\hat{i}$-2$\hat{j}$+3$\hat{k}$)

CARTESIAN EQUATION:

The cartesian equation of the line is,

x-x₁ /x₂ ₋ x₁ = y-y₁/y₂-y₁=z-z₁/z₂-z₁  --------------(2)

here , (x₁ ,y₁ ,z₁) = (4,7,1)

         (x₂ ,y₂, z₂) = (3,5,3)

By substituting the values of (x₁ ,y₁ ,z₁) & (x₂ ,y₂, z₂) in equation (2),we get

x-4/3-4 = y-7/5-7 = z-1/3-1

  x-4/-1 =  y-7/-2  =z-1/2

The vector equation is $\overrightarrow{r}$=4$\hat{i}$+7$\hat{j}$+$\hat{k}$+λ(-$\hat{i}$-2$\hat{j}$+3$\hat{k}$)

The cartesian equation is x-4/-1 =  y-7/-2  =z-1/2