Q.Find the equation of a line parallel to x-axis and passing through the origin.

The key idea is that a line parallel to the x-axis has a direction vector along the x-axis, and passing through the origin means the position vector of a point on the line is simply a scalar multiple of that direction vector.

Step 1: The direction vector of the x-axis is $\hat{i}$ (or any scalar multiple like $1\hat{i} + 0\hat{j} + 0\hat{k}$).

Step 2: The line passes through the origin, so the position vector of the origin is $\vec{0}$.

Step 3: The vector equation of a line is $\vec{r} = \vec{a} + \lambda \vec{b}$, where $\vec{a}$ is a point on the line and $\vec{b}$ is the direction vector. Substituting $\vec{a} = \vec{0}$ and $\vec{b} = \hat{i}$ gives $\vec{r} = \lambda \hat{i}$.

The key idea is simple: a line parallel to the x‑axis can only move along the x‑direction — it never changes its y or z coordinates. So every point on the line has the same y and the same z. Since the line also passes through the origin, those constant values are zero.

In three‑dimensional geometry, the vector equation of a line is $\vec{r} = \vec{a} + \lambda \vec{b}$, where $\vec{a}$ is the position vector of a fixed point on the line and $\vec{b}$ is a vector along the line (the direction vector). For a line parallel to the x‑axis, the direction vector must be parallel to $\hat{i}$, i.e. $(1,0,0)$. And “passing through the origin” means $\vec{a} = \vec{0}$.

Let’s build it step by step.

1. Direction vector of the line

   A line parallel to the x‑axis has the same direction as the x‑axis. The unit vector along the x‑axis is $\hat{i} = (1,0,0)$. So we can take the direction vector $\vec{b} = \hat{i}$ (or any scalar multiple, like $2\hat{i}$ — they all give the same line).

   Tip

   Any vector of the form $(k,0,0)$ with $k \neq 0$ works as a direction vector. Using $(1,0,0)$ is simplest.

2. Fixed point on the line

   The line passes through the origin $O(0,0,0)$. Its position vector is $\vec{a} = \vec{0} = 0\hat{i} + 0\hat{j} + 0\hat{k}$.

3. Vector equation

   Substitute into $\vec{r} = \vec{a} + \lambda \vec{b}$:

That’s the vector equation. In component form, $\vec{r} = (x,y,z) = (\lambda, 0, 0)$.

4. Cartesian equations From the component form, we read off:

Since $\lambda$ is a free parameter, $x$ can be any real number. The conditions $y=0$ and $z=0$ are the equations that describe the line in Cartesian form.

5. Why this makes sense Every point on the line has coordinates $(\lambda, 0, 0)$. As $\lambda$ varies over all real numbers, we get every point on the x‑axis. The line is exactly the x‑axis itself. So “parallel to the x‑axis and passing through the origin” is just the x‑axis.

Method: Equation of a Line Parallel to a Coordinate Axis

Use this when a line must be parallel to one of the axes (or to any fixed direction) and pass through a known point.

Steps

Step 1: Read off the direction vector from the "parallel to" clause.

Parallel to the x-axis means direction $\hat{i}=(1,0,0)$; parallel to the y-axis means $\hat{j}=(0,1,0)$; parallel to the z-axis means $\hat{k}=(0,0,1)$. Any non-zero multiple works equally well.

Step 2: Read off the fixed point.

The point the line passes through gives the position vector $\vec{a}$. "Through the origin" means $\vec{a}=\vec{0}$.

Step 3: Assemble $\vec{r}=\vec{a}+\lambda\vec{b}$ and, if asked, split into Cartesian form.

Substitute the point and direction into $\vec{r}=\vec{a}+\lambda\vec{b}$. To get Cartesian equations, equate components: the coordinate along the axis becomes the free parameter, and the other two coordinates are fixed constants. Remember a line in 3D needs two Cartesian equations (e.g. $y=0$ and $z=0$ for the x-axis), not one.

Common Mistakes

Mistake 1: Giving only one Cartesian equation for the line.

Why it's wrong: a line in 3D is the intersection of two planes, so it needs two equations; the x-axis is $y=0$ AND $z=0$, not just one of them. Correct approach: state both $y=0$ and $z=0$ alongside $\vec{r}=\lambda\hat{i}$.

Mistake 2: Using the wrong direction vector for "parallel to the x-axis".

Why it's wrong: parallel to the x-axis means the direction is $\hat{i}=(1,0,0)$, not $(0,1,0)$ or $(1,1,0)$. Correct approach: take $\vec{b}=\hat{i}$ and, since it passes through the origin, $\vec{a}=\vec{0}$.