Equation of y-axis in space, in vector form is :

The correct answer is Option (4) → \(\vec{r} = (0,0,0)+\lambda(0,1,0)\)

Note: The given answer is as per the official NTA Answer Key. However, more than one answer is possible: Option (2) and Option (4) are both correct, as explained below.

A line \(\vec{r} = \vec{a} + \lambda\vec{b}\) passes through the point \(\vec{a}\) and has direction \(\vec{b}\). The y-axis is the set of points (0, y, 0): it passes through the origin, and also through (0, 1, 0), and its direction is (0, 1, 0). So a vector equation of the y-axis needs a point with x = 0 and z = 0, and a direction parallel to (0, 1, 0).

Option (1) \(\vec{r} = (0,0,0)+\lambda(1,0,0)\) — Incorrect. The direction (1, 0, 0) is along the x-axis, so this is the x-axis.

Option (2) \(\vec{r} = (0,1,0)+\mu(0,1,0)\) — Should also be correct, though not the answer given by NTA. Its points are (0, 1 + μ, 0); as μ takes all real values, 1 + μ takes every real value, so these are exactly the points of the y-axis.

Option (3) \(\vec{r} = (0,1,0)+\mu(1,1,1)\) — Incorrect. For μ = 1 it gives (1, 2, 1), which is not on the y-axis.

Option (4) \(\vec{r} = (0,0,0)+\lambda(0,1,0)\) — Correct. Its points are (0, λ, 0): the line through the origin along the y-axis.

Advice for the exam: When more than one option is correct, mark the one in the simplest standard form, here Option (4), the line through the origin along (0, 1, 0). If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window, giving proper reasoning. NTA does revise its key when a challenge is well justified, so a sound challenge can raise your marks.