Question

If a line passes through the points $A(-1,0,2)$ and $B(3,4,6),$ then vector equation of the line is

[2 marks]

• A. $\overrightarrow{r}=-\hat{i}-2\hat{k}+\lambda (4\hat{i}+6\hat{j}+4\hat{k})$ where $\lambda \in R$
• B. $\overrightarrow{r}=-\hat{i}+2\hat{k}+\lambda (4\hat{i}+4\hat{j}+4\hat{k})$ where $\lambda \in R$
• C. $\overrightarrow{r}=-\hat{i}-2\hat{k}+\lambda (4\hat{i}-4\hat{j}+4\hat{k})$ where $\lambda \in R$
• D. $\overrightarrow{r}=\hat{i}+2\hat{k}+\lambda (4\hat{i}+4\hat{j}+4\hat{k})$ where $\lambda \in R$

Answer 1

The correct option is B $\overrightarrow{r}=-\hat{i}+2\hat{k}+\lambda (4\hat{i}+4\hat{j}+4\hat{k})$ where $\lambda \in R$

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be the position vectors of the points $A(-1,0,2)$ and $B(3,4,6).$
Then, $\overrightarrow{a}=-\hat{i}+2\hat{k}$ and $\overrightarrow{b}=3\hat{i}+4\hat{j}+6\hat{k}$
$\Rightarrow \overrightarrow{b}-\overrightarrow{a}=4\hat{i}+4\hat{j}+4\hat{k}$
Let $\overrightarrow{r}$ be the position vector of any arbitrary point on the line. Then vector equation of the line is
$\overrightarrow{r}=\overrightarrow{a}+\lambda (\overrightarrow{b}-\overrightarrow{a})$
$\Rightarrow \overrightarrow{r}=-\hat{i}+2\hat{k}+\lambda (4\hat{i}+4\hat{j}+4\hat{k})$