Question

Write the equation of the line passing through $(3,5,1)$ & which is parallel to the line formed by joining points $(2,5,7)$ & $(3,4,9)$.

• A. $\overrightarrow{r}=(-3,-5,-1)+\lambda (-1,1,-2)$
• B. $\overrightarrow{r}=(1,-1,2)+\lambda (3,5,1)$
• C. $\overrightarrow{r}=(3,5,1)+\lambda (1,-1,2)$
  
• D. $\overrightarrow{r}=(-1,1,-2)+\lambda (-3,-5,-1)$

Answer 1

The correct option is C

$\overrightarrow{r}=(3,5,1)+\lambda (1,-1,2)$

The line is parallel to the line formed by joining points $A(2,5,7)$ and $B(3,4,9)$

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ are the position vectors of points $A$ and $B$

then

$\overrightarrow{a}=2\hat{i}+5\hat{j}+7\hat{k}$

$\overrightarrow{b}=3\hat{i}+4\hat{j}+9\hat{k}$

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

Now, the line passes through the point $(3,5,1)$ and parallel to the vector

$\hat{i}-\hat{j}+2\hat{k}$.

So, the position vector $\overrightarrow{r}$ of any point on the line is,

$\overrightarrow{r}=3\hat{i}+5\hat{j}+\hat{k}+\lambda (\hat{i}-\hat{j}+2\hat{k})$