Question

The vector form of the equation of the line passing through points $(3,4,7)$ and $(5,1,6)$ is-

• A. $\overrightarrow{r}=(3\hat{i}+4\hat{j}-7\hat{k})+\lambda (2\hat{i}-3\hat{j}+13\hat{k})$
• B. $\overrightarrow{r}=(3\hat{i}+4\hat{j}-7\hat{k})+\lambda (8\hat{i}+5\hat{j}-\hat{k})$
• C. $\overrightarrow{r}=(3\hat{i}+4\hat{j}+7\hat{k})+\lambda (2\hat{i}-3\hat{j}-\hat{k})$
• D. $\overrightarrow{r}=(3\hat{i}+4\hat{j}-7\hat{k})+\lambda (2\hat{i}-3\hat{j}-13\hat{k})$

Answer 1

The correct option is C $\overrightarrow{r}=(3\hat{i}+4\hat{j}+7\hat{k})+\lambda (2\hat{i}-3\hat{j}-\hat{k})$

We have,

Given points are $(3,4,7)and(5,1,6)$.

We know that,

Vector equation of a line passing through the two points with position vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is

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

Let the given points are $A(3,4,7)and(5,1,6)$

Now,

Equation of vector,

$\overrightarrow{r}=(3\overrightarrow{i}+4\overrightarrow{j}+7\overrightarrow{k})+\lambda (5\overrightarrow{i}+\overrightarrow{j}+6\overrightarrow{k}-3\overrightarrow{i}-4\overrightarrow{j}-7\overrightarrow{k})$

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

Hence, this is the answer.