Question

The point having position vector \(4\hat{i}-11\hat{j}+2\hat{k}\) lies on the line

• A. \(\overset{̄}{r} = (6\hat{i}-4\hat{j}+5\hat{k})+λ(\hat{i}+7\hat{j}+3\hat{k})\)
• B. \(\overset{̄}{r} = (6\hat{i}-4\hat{j}+5\hat{k})+λ(2\hat{i}+\hat{j}+3\hat{k})\)
• C. \(\overset{̄}{r} = (6\hat{i}-4\hat{j}+5\hat{k})+λ(2\hat{i}+3\hat{j}+\hat{k})\)
• D. \(\overset{̄}{r} = (6\hat{i}-4\hat{j}+5\hat{k})+λ(2\hat{i}+7\hat{j}+3\hat{k})\)

Hint:

Subtract the base point of each line from the given point and check whether the difference is a multiple of the direction vector.

Answer 1

The Correct Option is D

Step 1: Substitute the point.
Write $4 = 6 + 2\lambda$ so $\lambda = -1$. Then check the other two coordinates for the direction of (D).

Step 2: Option D.
$y = -4 + 7(-1) = -11$ and $z = 5 + 3(-1) = 2$. All three coordinates agree.

Step 3: Other options.
For (A): $\lambda = -2$ from $x$, but then $y = -4 - 14 = -18 \neq -11$. For (B): $\lambda = -1$ gives $y = -5$. For (C): $\lambda = -1$ gives $y = -7$. All fail.

Final Answer:
Option (D). \[ \boxed{\text{(D)}} \]