Line through a point and parallel to a vector (vector form)
Line through a point and parallel to a vector (vector form)
In 3D geometry, a line can be defined in various ways, one of which is using the vector form. When we want to define a line that passes through a specific point and is parallel to a given vector, we use the vector equation of a line.
Vector Equation of a Line
The vector equation of a line in three-dimensional space is given by:
$$ \vec{r} = \vec{a} + \lambda \vec{b} $$
where:
- $\vec{r}$ is the position vector of any point on the line,
- $\vec{a}$ is the position vector of a fixed point through which the line passes (known point),
- $\vec{b}$ is the direction vector parallel to the line (given vector),
- $\lambda$ is a scalar parameter (real number).
Understanding the Components
Position Vector of a Known Point ($\vec{a}$)
This is the vector that represents the coordinates of the point through which the line passes. It is often given in the form $\vec{a} = \langle a_1, a_2, a_3 \rangle$, where $a_1$, $a_2$, and $a_3$ are the x, y, and z coordinates of the point, respectively.
Direction Vector ($\vec{b}$)
The direction vector is parallel to the line and dictates the orientation of the line in space. It is not unique; any non-zero scalar multiple of a direction vector is also a direction vector for the same line.
Scalar Parameter ($\lambda$)
The scalar parameter, often denoted by $\lambda$, allows us to find any point on the line by varying its value. For each value of $\lambda$, there is a corresponding point on the line.
Differences and Important Points
| Aspect | Description |
|---|---|
| Known Point ($\vec{a}$) | The fixed point through which the line passes. It anchors the line in space. |
| Direction Vector ($\vec{b}$) | Determines the direction of the line. It is essential for the line's orientation. |
| Scalar Parameter ($\lambda$) | Allows for the traversal along the line to find different points. It is not fixed. |
Formulas
The vector equation of a line can also be written in component form as:
$$ \vec{r} = \langle x, y, z \rangle $$ $$ \vec{a} = \langle a_1, a_2, a_3 \rangle $$ $$ \vec{b} = \langle b_1, b_2, b_3 \rangle $$
Then the equation becomes:
$$ \langle x, y, z \rangle = \langle a_1, a_2, a_3 \rangle + \lambda \langle b_1, b_2, b_3 \rangle $$
Which can be broken down into the parametric equations of the line:
$$ x = a_1 + \lambda b_1 $$ $$ y = a_2 + \lambda b_2 $$ $$ z = a_3 + \lambda b_3 $$
Examples
Example 1: Basic Line Equation
Suppose we have a line that passes through the point $P(1, 2, 3)$ and is parallel to the vector $\vec{b} = \langle 4, 5, 6 \rangle$. The vector equation of the line is:
$$ \vec{r} = \langle 1, 2, 3 \rangle + \lambda \langle 4, 5, 6 \rangle $$
Example 2: Finding a Point on the Line
Using the line from Example 1, find the coordinates of the point on the line when $\lambda = 2$.
$$ \vec{r} = \langle 1, 2, 3 \rangle + 2 \langle 4, 5, 6 \rangle $$ $$ \vec{r} = \langle 1 + 8, 2 + 10, 3 + 12 \rangle $$ $$ \vec{r} = \langle 9, 12, 15 \rangle $$
So, the coordinates of the point are $(9, 12, 15)$.
Example 3: Parallel Lines
If we have another line that passes through the point $Q(0, -1, 2)$ and is parallel to the same vector $\vec{b} = \langle 4, 5, 6 \rangle$, its equation is:
$$ \vec{r} = \langle 0, -1, 2 \rangle + \mu \langle 4, 5, 6 \rangle $$
Note that even though the lines are parallel and have the same direction vector, they are distinct because they pass through different points.
In conclusion, the vector form of a line is a powerful way to represent lines in 3D space. It is essential to understand the role of each component in the equation to manipulate and interpret lines in various geometric problems.