Line through a Point and Parallel to a Vector (Cartesian Form)

In 3D geometry, a line can be defined in various ways. One common method is to specify a line by giving a point through which the line passes and a direction vector that indicates the line's direction. This is known as the vector equation of a line. In this content, we will explore how to represent a line in Cartesian form when we know a point on the line and a vector parallel to the line.

Vector Equation of a Line

The vector equation of a line in three-dimensional space can be written as:

$$ \vec{r} = \vec{a} + t\vec{b} $$

where:

• $\vec{r}$ is the position vector of a generic point on the line,
• $\vec{a}$ is the position vector of a known point through which the line passes,
• $\vec{b}$ is a direction vector parallel to the line,
• $t$ is a scalar parameter.

Cartesian Equation of a Line

The Cartesian equation of a line is derived from the vector equation by equating the components. If the direction vector $\vec{b}$ has components $(b_1, b_2, b_3)$ and the known point has coordinates $(x_0, y_0, z_0)$, then the Cartesian form of the line is:

$$ \frac{x - x_0}{b_1} = \frac{y - y_0}{b_2} = \frac{z - z_0}{b_3} $$

This assumes that $b_1, b_2,$ and $b_3$ are all non-zero. If any of the components of $\vec{b}$ is zero, the corresponding fraction is replaced by the equation of the coordinate that is constant.

Differences and Important Points

Aspect	Vector Equation	Cartesian Equation
Representation	Parametric form	Non-parametric form
Components	Requires a direction vector	Requires the direction ratios
Known Point	Position vector of the known point	Coordinates of the known point
Parameter	Scalar parameter $t$	No explicit parameter
Applicability	General use in vector calculus	Convenient for intersection and parallelism
Ease of Visualization	Less intuitive	More intuitive for those familiar with graphs

Formulas

• Vector Equation: $\vec{r} = \vec{a} + t\vec{b}$
• Cartesian Equation: $\frac{x - x_0}{b_1} = \frac{y - y_0}{b_2} = \frac{z - z_0}{b_3}$

Examples

Example 1: Vector Equation to Cartesian Form

Given a line passing through the point $P(1, 2, 3)$ and parallel to the vector $\vec{b} = \begin{pmatrix} 2 \ -1 \ 4 \end{pmatrix}$, find the Cartesian equation of the line.

Solution:

The position vector of point $P$ is $\vec{a} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}$.

The vector equation of the line is:

$$ \vec{r} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} + t\begin{pmatrix} 2 \ -1 \ 4 \end{pmatrix} $$

To find the Cartesian equation, we equate the components:

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

This is the Cartesian form of the line.

Example 2: Cartesian Equation with a Zero Component

Find the Cartesian equation of a line passing through the point $Q(3, -1, 5)$ and parallel to the vector $\vec{c} = \begin{pmatrix} 0 \ 3 \ -2 \end{pmatrix}$.

Solution:

Since the $x$-component of $\vec{c}$ is zero, the line is parallel to the $yz$-plane, and the $x$-coordinate of any point on the line is constant ($x = 3$).

The Cartesian equation is:

$$ x = 3, \quad \frac{y + 1}{3} = \frac{z - 5}{-2} $$

This represents the same line in Cartesian form.

In conclusion, converting a line from vector form to Cartesian form involves using the known point and direction vector to set up a relationship between the coordinates of any point on the line. This method is widely used in 3D geometry problems, especially when dealing with intersections, parallelism, and distances.