Question

Find the vector equation of the line through A (3,4,-7) and B(1,-1,6). ​

Answer 1

Answer:

$\displaystyle \vec{r}=(3\hat{i}+4\hat{j}-7\hat{k})+\lambda(-2\hat{i}-5\hat{j}+13\hat{k})$

Explanation:

We know :

Vector equation of line passing through the points having position vectors :

$\displaystyle \vec{a} \ \text{and} \ \vec{b} \ \text{given by}$

$\displaystyle \vec{r}=\vec{a}+\lambda(\vec{b}-\vec{a})$

We have :

$\displaystyle \vec{a}=3\hat{i}+4\hat{j}-7\hat{k} \ \text{and}$

$\displaystyle \vec{b}=\hat{i}-\hat{j}+6\hat{k}$

So , vector equation of line passing through  A ( 3 , 4 , - 7 ) and B ( 1 , - 1 , 6 ) :

$\displaystyle \vec{r}=(3\hat{i}+4\hat{j}-7\hat{k})+\lambda[(\hat{i}-\hat{j}+6\hat{k})-(3\hat{i}+4\hat{j}-7\hat{k})]$

$\displaystyle \vec{r}=(3\hat{i}+4\hat{j}-7\hat{k})+\lambda(\hat{i}-\hat{j}+6\hat{k}-3\hat{i}-4\hat{j}+7\hat{k})$

$\displaystyle \vec{r}=(3\hat{i}+4\hat{j}-7\hat{k})+\lambda(-2\hat{i}-5\hat{j}+13\hat{k})$

Hence we get required answer.