The vector equation of the line (x-3)/2=(y+1)/(-4)=(z-5)/1 is

The correct answer is Option (3) → \(\vec{r} = (3\hat{\imath}-\hat{\jmath}+5\hat{k})+\lambda(2\hat{\imath}-4\hat{\jmath}+\hat{k})\)

The line \(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\) passes through \((x_1, y_1, z_1)\) and is parallel to \(a\hat{\imath}+b\hat{\jmath}+c\hat{k}\). Its vector form is \(\vec{r} = \vec{a}+\lambda\vec{b}\).

Here the point is (3, −1, 5), because \(y+1 = y-(-1)\):

    \(\vec{a} = 3\hat{\imath}-\hat{\jmath}+5\hat{k}\)

The direction comes from the denominators 2, −4, 1:

    \(\vec{b} = 2\hat{\imath}-4\hat{\jmath}+\hat{k}\)

So \(\vec{r} = (3\hat{\imath}-\hat{\jmath}+5\hat{k})+\lambda(2\hat{\imath}-4\hat{\jmath}+\hat{k})\).

Why the other options are incorrect

• Option (1) \(\vec{r} = (2\hat{\imath}-4\hat{\jmath}+\hat{k})+\lambda(3\hat{\imath}-\hat{\jmath}+5\hat{k})\) — Incorrect. The point and the direction have been swapped.

• Option (2) \(\vec{r} = (-3\hat{\imath}+\hat{\jmath}-5\hat{k})+\lambda(2\hat{\imath}-4\hat{\jmath}+\hat{k})\) — Incorrect. The signs of the point are reversed; (−3, 1, −5) does not satisfy the given equation.

• Option (4) \(\vec{r} = (3\hat{\imath}-\hat{\jmath}+5\hat{k})+\lambda(2\hat{\imath}+4\hat{\jmath}+\hat{k})\) — Incorrect. The y-direction ratio is −4, not 4.