Vector Equation of a Line
The vector equation of a line is a fundamental concept in analytical geometry, providing a concise way to describe lines in both two-dimensional and three-dimensional space. This equation is expressed as:
$$ \mathbf{r} = \mathbf{a} + \lambda\mathbf{b} $$
Where:
- $\mathbf{r}$ represents any point on the line
- $\mathbf{a}$ is a position vector of a known point on the line
- $\mathbf{b}$ is a direction vector parallel to the line
- $\lambda$ is a scalar parameter
The vector equation of a line essentially describes the line as a set of points that can be reached by starting at a fixed point $\mathbf{a}$ and moving in the direction of $\mathbf{b}$ for some distance determined by $\lambda$.
Components of the Vector Equation
Position Vector $\mathbf{a}$
- The position vector $\mathbf{a}$ represents a known point on the line.
- It anchors the line in space, giving us a reference point from which to describe all other points on the line.
In 3D space, if we know the line passes through the point (1, 2, 3), then
\[\mathbf{a} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \]
Direction Vector $\mathbf{b}$
- The direction vector $\mathbf{b}$ indicates the direction in which the line extends.
- It's parallel to the line and determines its orientation in space.
If the line is parallel to the vector \(\mathbf{a} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} \), then we can use this as our direction vector $\mathbf{b}$.
Any scalar multiple of $\mathbf{b}$ can be used as the direction vector, as it only affects the parameterization of the line, not its geometric properties.
Scalar Parameter $\lambda$
- The scalar $\lambda$ (lambda) acts as a parameter that allows us to generate all points on the line.
- As $\lambda$ varies over all real numbers, $\mathbf{r}$ traces out the entire line.
Students often confuse $\lambda$ with a fixed value. Remember, $\lambda$ can take any real value, each corresponding to a different point on the line.
Forms of Line Equations
Parametric Form
The parametric form of a line equation is derived directly from the vector equation. In three dimensions, it's written as:
