Question

The vector equation of the line joining the points (2, 1, 3) and (-2, 4, 1) is

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

Answer 1

The Correct Option is A

Given: Points \( A(2, 1, 3) \) and \( B(-2, 4, 1) \) 

Step 1: Find the direction vector

The direction vector is given by:

\[ \overrightarrow{AB} = B - A = (-2 - 2, 4 - 1, 1 - 3) = (-4, 3, -2) \]

Step 2: Write the vector equation of the line

The vector equation of the line passing through point \( A(2, 1, 3) \) with direction vector \( \overrightarrow{AB} = (-4, 3, -2) \) is:

\[ \overrightarrow{r} = \overrightarrow{A} + \lambda \overrightarrow{AB} \] \[ \overrightarrow{r} = 2\hat{i} + \hat{j} + 3\hat{k} + \lambda(-4\hat{i} + 3\hat{j} - 2\hat{k}) \]

Final Answer:

\[ \overrightarrow{r} = 2\hat{i} + \hat{j} + 3\hat{k} + \lambda(-4\hat{i} + 3\hat{j} - 2\hat{k}) \]