The vector equation of the line passing through points \(A(3, 4, -7)\) and \(B(1, -1, 6)\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\vec{x} = (3 - 2\lambda)\hat{\imath} + (4 - 5\lambda)\hat{\jmath} + (-7 + 13\lambda)\hat{k}\)
The line through A and B has direction \(\vec b - \vec a\), where \(\vec a\) and \(\vec b\) are the position vectors of A and B:
\(\vec b - \vec a = (1 - 3)\hat{\imath} + (-1 - 4)\hat{\jmath} + (6 + 7)\hat{k} = -2\hat{\imath} - 5\hat{\jmath} + 13\hat{k}\)
The vector equation \(\vec{x} = \vec a + \lambda(\vec b - \vec a)\) gives
\(\vec{x} = 3\hat{\imath} + 4\hat{\jmath} - 7\hat{k} + \lambda(-2\hat{\imath} - 5\hat{\jmath} + 13\hat{k})\)
\(\vec{x} = (3 - 2\lambda)\hat{\imath} + (4 - 5\lambda)\hat{\jmath} + (-7 + 13\lambda)\hat{k}\)
As a check, \(\lambda = 0\) gives A and \(\lambda = 1\) gives B.
Why the other options are incorrect
Option (1) \(\vec{x} = 3\hat{\imath} + 4\hat{\jmath} - 7\hat{k} + \lambda(\hat{\imath} - \hat{\jmath} + 6\hat{k})\) — Incorrect. It uses the position vector of B as the direction. B is not on this line: x = 1 needs \(\lambda = -2\), which gives y = 6, not −1.
Option (3) \(\vec{x} = \hat{\imath} - \hat{\jmath} + 6\hat{k} + \lambda(3\hat{\imath} + 4\hat{\jmath} - 7\hat{k})\) — Incorrect. It uses the position vector of A as the direction. A is not on this line: x = 3 needs \(\lambda = \frac{2}{3}\), which gives \(y = \frac{5}{3}\), not 4.
Option (4) \(\vec{x} = (-2 + 3\lambda)\hat{\imath} + (-5 + 4\lambda)\hat{\jmath} + (13 - 7\lambda)\hat{k}\) — Incorrect. It passes through B (\(\lambda = 1\)) but not through A: x = 3 needs \(\lambda = \frac{5}{3}\), which gives \(y = \frac{5}{3}\), not 4.