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: Understanding the Concept:
A point \(P\) lies on the line \(\bar{r} = \bar{a} + \lambda\bar{d}\) if \(\overrightarrow{AP} = \bar{P} - \bar{a}\) is a scalar multiple of \(\bar{d}\).

Step 2: Compute the difference.
All four lines have the same base point \((6, -4, 5)\). The difference is \((4 - 6,\ -11 + 4,\ 2 - 5) = (-2, -7, -3)\).

Step 3: Compare with each direction.
\((-2, -7, -3) = -1\times(2, 7, 3)\). This matches the direction in option (D).

Step 4: Why the other options are wrong.
(A) \((1, 7, 3)\): ratios \(-2, -1, -1\) are not equal. (B) \((2, 1, 3)\): ratios \(-1, -7, -1\) are not equal. (C) \((2, 3, 1)\): ratios \(-1, -7/3, -3\) are not equal.

Final Answer:
The point lies on the line in option (D) with \(\lambda = -1\). \[ \boxed{\text{(D)}} \]