Straight Line in 3D — Equations

In two dimensions, a line is pinned down by a single equation: y = mx + c, or ax + by + c = 0. One equation, two unknowns, one degree of freedom — that is a line.

In three dimensions, something changes. You have three coordinates (x, y, z), and a single equation like x + y + z = 6 does not define a line — it defines a plane. To cut down from a 3D space to a 1D line, you need two equations, not one.

But there is a more natural way to think about a line that avoids needing two simultaneous equations. Instead of describing a line as the intersection of two surfaces, describe it the way you would actually draw it: pick a point, pick a direction, and walk.

That is the fundamental idea behind all three forms of the equation of a line in 3D. You give a starting point and a direction, and the equation traces out every point you visit as you walk along that direction — forward and backward, forever.

What you need: a point and a direction

To specify a line in 2D, a slope and a point were enough. In 3D, a "slope" is not a single number — a line can tilt in three independent directions at once. So instead of a slope, you need a direction vector or a set of direction ratios.

A line in 3D is uniquely determined by:

1. A point it passes through: P_0 = (x_1, y_1, z_1)
2. A direction: given by direction ratios (a, b, c), or equivalently a vector \vec{b} = a\hat{i} + b\hat{j} + c\hat{k}

Given these two pieces of information, you can write the equation of the line in three equivalent ways. Each has its own advantages.

Vector form

Let the position vector of the known point P_0 be \vec{a} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}, and let the direction vector be \vec{b} = a\hat{i} + b\hat{j} + c\hat{k}.

A general point P lies on the line if and only if the vector from P_0 to P is parallel to \vec{b}. Two vectors are parallel when one is a scalar multiple of the other. So \vec{P_0P} = \lambda\vec{b} for some real number \lambda.

If \vec{r} is the position vector of P, then \vec{P_0P} = \vec{r} - \vec{a}. The condition becomes:

Rearranging:

Reading it. Start at \vec{a} (the known point). Add \lambda copies of \vec{b} (the direction). When \lambda = 0, you are at P_0 itself. When \lambda = 1, you have moved one full direction-vector length forward. When \lambda = -1, you have moved one full length backward. As \lambda ranges over all real numbers, you trace out every point on the line.

This is the most compact form. It is one equation, it works in any number of dimensions, and it encodes the geometric idea directly: a line is a point plus a direction.

Cartesian (symmetric) form

The vector equation \vec{r} = \vec{a} + \lambda\vec{b} is really three equations hiding inside one. Write \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} and expand:

Match components:

Solve each for \lambda:

Since all three expressions equal the same \lambda, set them equal to each other:

This is called the symmetric form because all three variables appear in the same way — each coordinate minus its reference value, divided by the corresponding direction ratio.

What if a direction ratio is zero? If, say, a = 0, then the line has no component in the x-direction — it lies in a plane parallel to the yz-plane, with x constant. In this case, the symmetric form is written as

You cannot divide by zero, so you separate the zero-ratio coordinate into its own equation. This is not a special case of the algebra — it is the algebra telling you something geometric: the line does not move in that direction, so the coordinate is fixed.

Similarly, if two direction ratios are zero, say b = c = 0, the line is parallel to the x-axis, and the equations become y = y_1, z = z_1 — the line is the intersection of two planes.

Parametric form

Go back to the three component equations before you eliminated \lambda:

This is the most computational form. Given any value of \lambda, you can immediately read off the coordinates of the corresponding point. It is especially useful for:

• Finding where a line intersects a plane (substitute the parametric expressions into the plane equation and solve for \lambda).
• Computing specific points at known fractions along the line.
• Plotting the line — just tabulate (x, y, z) for several values of \lambda.

The three forms are not three different things. They are three notations for the same line. The vector form is the most elegant. The symmetric form is the most common in Indian textbooks and JEE problems. The parametric form is the most useful for computation. You should be able to move between them without thinking.

A line through two points

Often you are not given a direction vector — you are given two points P_1(x_1, y_1, z_1) and P_2(x_2, y_2, z_2) and asked to write the equation of the line through both.

The direction vector is simply \vec{P_1P_2} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}. The direction ratios are the coordinate differences: (x_2 - x_1, \; y_2 - y_1, \; z_2 - z_1).

Use either point as the "known point" — the resulting line is the same. In symmetric form:

In vector form:

where \vec{a_1} and \vec{a_2} are the position vectors of the two points.

Reading this. When \lambda = 0, \vec{r} = \vec{a_1} — you are at P_1. When \lambda = 1, \vec{r} = \vec{a_2} — you are at P_2. For 0 < \lambda < 1, you are between them. For \lambda > 1, you have gone past P_2. For \lambda < 0, you are on the other side of P_1. The parameter \lambda locates you on the line like a ruler.

Computing one from start to finish

Common confusions

• "The symmetric form is one equation." It looks like one chain, but \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} is actually two independent equations: \frac{x-x_1}{a} = \frac{y-y_1}{b} and \frac{y-y_1}{b} = \frac{z-z_1}{c}. Two equations in three unknowns — one degree of freedom — a line. Some problems ask you to "find the two planes whose intersection is this line." Those two planes come directly from splitting the chain.

• "You need direction cosines, not direction ratios." For writing the equation of a line, direction ratios are sufficient. Direction cosines are normalised direction ratios — they are useful for computing angles, but you do not need them to write or use the equation. In fact, using direction ratios keeps the numbers cleaner.

• "\lambda is always positive." No — \lambda ranges over all real numbers, including negative values and zero. Negative \lambda means you are on the opposite side of P_0 from the direction \vec{b}. Restricting \lambda would give a ray, not a line.

• "You must simplify direction ratios." You do not have to. Direction ratios (6, -6, -6) give the same line as (1, -1, -1) — the parameter \lambda absorbs the scaling. Simplification makes the numbers nicer but is not algebraically necessary.

• "The vector form and symmetric form give different lines." They never do. They are two notations for the same line. If you derive one from the other and get something different, there is an arithmetic error.

• "A line in 3D has a slope." In 2D, a line has a single slope m = \Delta y / \Delta x. In 3D, a line has three direction ratios (a, b, c), which you can think of as three "partial slopes." There is no single number that captures the direction of a line in 3D.

Going deeper

Why "symmetric" form?

The name "symmetric form" comes from the way the three coordinates appear symmetrically — each one has the same structure: (x - x_1)/a. This uniformity makes it easy to read off the point and direction at a glance, and it generalises naturally. The form is sometimes also called the "Cartesian form" or the "standard form" of a line in 3D.

In contrast, the parametric form breaks the symmetry by introducing \lambda explicitly. The vector form restores symmetry by hiding the components inside vector notation. All three are useful; the symmetric form is the one most commonly tested in Indian board exams and JEE.

Converting between forms — a recipe

Intersection with a coordinate plane

A common problem type: find where the line meets the xy-plane. The xy-plane is z = 0. Set z = 0 in the parametric equations:

Then substitute this \lambda back into the x and y parametric equations to get the intersection point. If c = 0, the line is parallel to the xy-plane and never meets it (unless z_1 = 0, in which case the entire line lies in the xy-plane).

The same technique works for any plane: substitute the parametric expressions into the plane equation, solve for \lambda, and recover the coordinates.

Checking if a point lies on a line

Given a point (p, q, s) and a line \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}, substitute the point:

If all three fractions are equal, the point lies on the line (and their common value is the parameter \lambda at that point). If any fraction differs from the others, the point is not on the line.

This is a fast membership test and is used frequently in problems that ask whether three points are collinear. Three points A, B, C are collinear if and only if C lies on the line through A and B.

Parallel and identical lines

Two lines are parallel if their direction ratios are proportional: \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}. Parallel lines either never meet (they are distinct) or they are the same line (they coincide everywhere).

To check whether two parallel lines are identical, take the known point of one line and test whether it lies on the other line using the fraction test above. If it does, the lines are identical. If not, they are parallel and distinct — in 3D, this means they could be coplanar (lying in a common plane, like railway tracks) but they could also be skew lines that share a direction but live on different parallel planes. Two lines that are parallel are always coplanar, so parallel distinct lines are never skew.

Skew lines — a 3D phenomenon

In 2D, two non-parallel lines always intersect. In 3D, this is no longer true. Two lines can be non-parallel and still never meet — they pass each other at different heights, like two overpasses on a highway interchange. Such lines are called skew lines. They have no point in common and are not parallel.

You will study skew lines, the shortest distance between them, and how to detect them in the article on angles and distances between lines.

The line as an intersection of two planes

The symmetric form \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} encodes two equations:

Cross-multiplying the first: b(x - x_1) = a(y - y_1), which is bx - ay = bx_1 - ay_1. This is the equation of a plane.

Cross-multiplying the second: c(y - y_1) = b(z - z_1), which is cy - bz = cy_1 - bz_1. This is the equation of another plane.

The line is the intersection of these two planes. Every line in 3D can be expressed as the intersection of two planes — this is the algebraic version of the geometric fact that two non-parallel planes in 3D meet in a line.

This perspective is useful when you need to find a plane containing a given line (the line's two planes give you a one-parameter family of planes containing it) or when you need to find where two lines intersect (solve four equations in three unknowns, and check for consistency).

Where this leads next

You can now write the equation of any line in 3D. The next natural questions are: what angle do two lines make? How far apart are two lines that do not meet? And what happens when lines and planes interact?

• Straight Line in 3D — Angles and Distances — the angle between two lines, the shortest distance between two skew lines, and the condition for two lines to intersect.
• Plane — Basic Equations — the equation of a plane in 3D, and where lines meet planes.
• Direction Cosines and Ratios — the foundation for the direction ratios used in line equations, if you want to revisit it.
• Straight Line Forms (2D) — the 2D version of line equations, for comparison. The slope-intercept form y = mx + c is a special case of the parametric form with the z-coordinate removed.
• Vectors Introduction — the vector language that makes the vector form of the line equation natural.