Line Passing Through Two Points (Cartesian Form)

In 3D geometry, the equation of a line can be determined if we know two distinct points through which the line passes. The Cartesian form of the equation of a line is derived using the coordinates of these two points.

Understanding the Concept

Given two points, $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, in a three-dimensional space, the line passing through these points can be represented in vector form and then converted into Cartesian form.

Vector Form

The vector equation of the line is given by:

$$ \vec{r} = \vec{r_1} + t(\vec{r_2} - \vec{r_1}) $$

where $\vec{r}$ is the position vector of a general point on the line, $\vec{r_1}$ and $\vec{r_2}$ are the position vectors of points $P_1$ and $P_2$ respectively, and $t$ is a scalar parameter.

Cartesian Form

To convert the vector equation to Cartesian form, we equate the corresponding components:

$$ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} = t $$

This is the Cartesian form of the equation of a line passing through two points in 3D space.

Important Points and Differences

Aspect	Description
Direction Ratios (DRs)	The differences $x_2 - x_1$, $y_2 - y_1$, and $z_2 - z_1$ are the direction ratios of the line.
Direction Cosines (DCs)	The direction cosines are the cosines of the angles that the line makes with the positive directions of the coordinate axes.
Symmetric Form	The Cartesian equation can also be written in symmetric form if the direction ratios are non-zero.

Formulas

The direction ratios (DRs) of the line are given by:

$$ DRs = (x_2 - x_1, y_2 - y_1, z_2 - z_1) $$

The direction cosines (DCs) are calculated using the direction ratios:

$$ DCs = \left( \frac{x_2 - x_1}{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}, \frac{y_2 - y_1}{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}, \frac{z_2 - z_1}{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}} \right) $$

Examples

Example 1: Find the Cartesian Equation

Given two points $A(1, 2, 3)$ and $B(4, 6, 5)$, find the Cartesian equation of the line passing through these points.

Solution:

First, we find the direction ratios (DRs):

$$ DRs = (4 - 1, 6 - 2, 5 - 3) = (3, 4, 2) $$

Now, we can write the Cartesian form:

$$ \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - 3}{2} $$

This is the required Cartesian equation of the line.

Example 2: Symmetric Form

If the direction ratios are non-zero, we can write the equation in symmetric form. For the same points $A(1, 2, 3)$ and $B(4, 6, 5)$:

Solution:

The symmetric form of the equation is:

$$ \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - 3}{2} $$

This can also be written as:

$$ \frac{x - 1}{4 - 1} = \frac{y - 2}{6 - 2} = \frac{z - 3}{5 - 3} $$

This is the symmetric form of the Cartesian equation of the line.

Understanding the Cartesian form of a line passing through two points is crucial for solving problems in 3D geometry. It allows us to find the equation of a line in a space given two points, which is a fundamental concept in vector algebra and analytical geometry.