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

The correct answer is Option (3) → \(\vec{r}=(-2\hat{\imath}-3\hat{\jmath}-4\hat{k})+\lambda(3\hat{\imath}+\hat{\jmath}+8\hat{k})\)

The line through points with position vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{r} = \vec{a} + \lambda(\vec{b}-\vec{a})\): a point on the line plus a multiple of the direction vector \(\vec{b}-\vec{a}\).

    \(\vec{a} = -2\hat{\imath}-3\hat{\jmath}-4\hat{k}\)
    \(\vec{b} = \hat{\imath}-2\hat{\jmath}+4\hat{k}\)

    \(\vec{b}-\vec{a} = (1+2)\hat{\imath}+(-2+3)\hat{\jmath}+(4+4)\hat{k}\)
    \(= 3\hat{\imath}+\hat{\jmath}+8\hat{k}\)

So

    \(\vec{r}=(-2\hat{\imath}-3\hat{\jmath}-4\hat{k})+\lambda(3\hat{\imath}+\hat{\jmath}+8\hat{k})\)

Why the other options are incorrect

• Option (1) \(\vec{r}=(-2\hat{\imath}-3\hat{\jmath}-4\hat{k})+\lambda(\hat{\imath}-2\hat{\jmath}+4\hat{k})\) — Incorrect. Its direction is the position vector of (1, −2, 4) itself, not \(\vec{b}-\vec{a}\); the line does not reach (1, −2, 4).

• Option (2) \(\vec{r}=(2\hat{\imath}+3\hat{\jmath}+4\hat{k})+\lambda(3\hat{\imath}-\hat{\jmath}+8\hat{k})\) — Incorrect. Its point is (2, 3, 4), not a given point, and its direction has −1 in place of +1 for \(\hat{\jmath}\).

• Option (4) \(\vec{r}=(2\hat{\imath}+3\hat{\jmath}+4\hat{k})+\lambda(3\hat{\imath}+\hat{\jmath}+8\hat{k})\) — Incorrect. The direction is right, but the point (2, 3, 4) is not on the line: 2 + 2 = 3λ gives λ = 4/3, while 3 + 3 = λ gives λ = 6.