Line Passing Through Two Points (Vector Form)

In 3D geometry, a line can be uniquely determined by two distinct points through which it passes. The vector form of the equation of a line provides a way to describe this line using vectors. This form is particularly useful in physics and engineering, where vectors are a common tool for describing directions and magnitudes.

Vector Representation

A vector in three-dimensional space can be represented as:

$$ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} $$

where $\hat{i}$, $\hat{j}$, and $\hat{k}$ are the unit vectors along the x, y, and z axes, respectively, and $x$, $y$, and $z$ are the coordinates of the vector.

Equation of a Line Through Two Points

Consider two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in space. The position vectors of these points are $\vec{a}$ and $\vec{b}$, respectively. The vector form of the equation of a line passing through these two points is given by:

$$ \vec{r} = \vec{a} + \lambda (\vec{b} - \vec{a}) $$

where $\vec{r}$ is the position vector of any point on the line, and $\lambda$ is a scalar parameter.

Important Points and Differences

Aspect	Description
Position Vector	A vector that represents the position of a point in space relative to the origin.
Direction Vector	A vector that gives the direction of the line. In the case of a line through two points, it is $\vec{b} - \vec{a}$.
Scalar Parameter	A real number ($\lambda$) that can be varied to get different points on the line.
Uniqueness	A line in 3D space is uniquely determined by two distinct points.
Infinite Points	The line contains an infinite number of points, which can be obtained by varying $\lambda$.

Formulas

Given two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, the vector form of the line is:

$$ \vec{r} = (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}) + \lambda [(x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}] $$

Examples

Example 1

Find the vector equation of the line passing through the points $P_1(1, 2, 3)$ and $P_2(4, 5, 6)$.

Solution:

The position vectors of $P_1$ and $P_2$ are $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = 4\hat{i} + 5\hat{j} + 6\hat{k}$, respectively.

The direction vector is $\vec{b} - \vec{a} = (4 - 1)\hat{i} + (5 - 2)\hat{j} + (6 - 3)\hat{k} = 3\hat{i} + 3\hat{j} + 3\hat{k}$.

The vector equation of the line is:

$$ \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda (3\hat{i} + 3\hat{j} + 3\hat{k}) $$

Example 2

Determine a point on the line in Example 1 when $\lambda = 2$.

Solution:

Substitute $\lambda = 2$ into the vector equation:

$$ \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + 2 (3\hat{i} + 3\hat{j} + 3\hat{k}) $$

$$ \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} + 6\hat{i} + 6\hat{j} + 6\hat{k} $$

$$ \vec{r} = 7\hat{i} + 8\hat{j} + 9\hat{k} $$

Thus, the point on the line when $\lambda = 2$ is $(7, 8, 9)$.

Conclusion

The vector form of the equation of a line passing through two points is a powerful tool in 3D geometry. It allows for the description of a line in terms of vectors, which can be easily manipulated and visualized. Understanding this concept is essential for solving problems in various fields, including physics, engineering, and computer graphics.