Equations of a Line in 3D

1. The Ingredients of a 3D Line

Just like in 2D coordinate geometry, a line in 3D space is uniquely determined by two pieces of information:

1. A fixed point that the line passes through.
2. The direction of the line.

In 3D, we specify the direction using a vector that is parallel to the line. The components of this parallel vector are called Direction Ratios (DRs). If we are given two points on the line, say $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, the direction ratios of the line are simply the components of the vector $\overrightarrow{AB}$, which are $(x_2-x_1,y_2-y_1,z_2-z_1)$.

For a quick and exam-oriented refresher on Direction Ratios and their cousins, Direction Cosines (DCs), the following video is excellent.

2. The Vector Equation of a Line

The vector equation is an elegant way to describe a line using position vectors. Imagine a line in space that passes through a point P and is parallel to a direction vector $\vec{b}$.

Let the position vector of the known point P be $\vec{a}$. Now, let's take any other arbitrary point X on the line, with position vector $\vec{r}$. The vector connecting P to X, which is $\overrightarrow{PX}$, must lie along the line. Since the line is parallel to the direction vector $\vec{b}$, the vector $\overrightarrow{PX}$ must be a scalar multiple of $\vec{b}$.

We can write this relationship as:

where $\lambda$ is a scalar parameter.

From the triangle law of vector addition on $△OPX$, we have $\overrightarrow{OP}+\overrightarrow{PX}=\overrightarrow{OX}$. Substituting our known vectors:

This gives us the standard vector equation of a line:

Here, $\vec{a}$ is the position vector of a point on the line, and $\vec{b}$ is the direction vector.

To see this derived and applied to an example, the following video provides a very clear, step-by-step explanation.

3. The Cartesian Equation of a Line

The Cartesian equation expresses the same relationship using the x, y, and z coordinates. It can be derived directly from the vector equation.

Let the known point be $P(x_1,y_1,z_1)$, so its position vector is $\vec{a}=x_1\hat{i}+y_1\hat{j}+z_1\hat{k}$.
Let the direction ratios of the line be $(a,b,c)$, so the direction vector is $\vec{b}=a\hat{i}+b\hat{j}+c\hat{k}$.
Let the general point on the line be $X(x,y,z)$, so its position vector is $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$.

Substituting these into the vector equation $\vec{r}=\vec{a}+\lambda \vec{b}$:

By comparing the coefficients of $\hat{i}$, $\hat{j}$, and $\hat{k}$, we get the parametric equations:

• $x=x_1+\lambda a$
• $y=y_1+\lambda b$
• $z=z_1+\lambda c$

If we solve each of these for the parameter $\lambda$, we get:

Since $\lambda$ is the same in all three, we can set them equal to each other to get the Cartesian (or symmetric) equation of the line:

4. The Two Fundamental Cases

All problems on finding the equation of a line boil down to one of two scenarios. The GeeksForGeeks article below is a good reference for both, presented in a clear, structured way.

5. Exam Focus: Conversions and Non-Standard Forms

For competitive exams like the MHT-CET, a common task is to convert between Cartesian and vector forms. More importantly, questions often present the Cartesian equation in a "non-pure" form, with coefficients on x, y, or z, or with terms rearranged. You must first convert it to the standard symmetric form before you can correctly identify the point and the direction ratios.

The following video is essential viewing as it specifically addresses these exam-oriented tricks.

Mastering this manipulation is key. Always ensure the coefficients of x, y, and z are +1 before reading the direction ratios from the denominators.

Conclusion

In this lesson, we have built a solid foundation for describing lines in three dimensions. You now have the tools to define a line and switch between its different mathematical representations.

Here are your key takeaways:

• A line in 3D is defined by a point on the line and a parallel direction vector.
• The vector equation is $\vec{r}=\vec{a}+\lambda \vec{b}$, where $\vec{a}$ is the position vector of the point and $\vec{b}$ is the direction vector.
• The Cartesian equation is $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$, where $(x_1,y_1,z_1)$ is the point and $(a,b,c)$ are the direction ratios.
• For exams, always convert non-standard Cartesian equations to the pure symmetric form before extracting the point or direction ratios.

In our next lesson, we will expand our study of 3D geometry by moving from one-dimensional lines to two-dimensional surfaces: planes. We will discover that they, too, have distinct vector and Cartesian equations.