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:
$$ x = x_0 + \lambda l $$
$$ y = y_0 + \lambda m $$
$$ z = z_0 + \lambda n $$
Where $(x_0, y_0, z_0)$ is a point on the line (corresponding to $\mathbf{a}$), and $(l, m, n)$ are the components of the direction vector $\mathbf{b}$.
ExampleFor a line passing through (1, 2, 3) in the direction of \( \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} \), the parametric equations are:
\[\begin{cases} x = 1 + 2\lambda\\ y = 2 - \lambda\\ z = 3 + 3\lambda\end{cases} \]
Cartesian Form
The Cartesian form of a line equation can be derived from the parametric form by eliminating the parameter $\lambda$. In three dimensions, it's expressed as:
$$ \frac{x-x_0}{l} = \frac{y-y_0}{m} = \frac{z-z_0}{n} $$
NoteThis form is particularly useful when we need to find the intersection of a line with a plane or another line.
ExampleUsing the same line as before, the Cartesian form would be:
$\frac{x-1}{2} = \frac{y-2}{-1} = \frac{z-3}{3}$
Angle Between Two Lines
- The angle between two lines can be calculated using the scalar product of their direction vectors.
- If $\mathbf{b}_1$ and $\mathbf{b}_2$ are the direction vectors of two lines, the angle $\theta$ between them is given by:
$$ \cos \theta = \frac{\mathbf{b}_1 \cdot \mathbf{b}_2}{|\mathbf{b}_1||\mathbf{b}_2|} $$
ExampleConsider two lines with direction vectors \[\mathbf{b_1} = \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} \quad\text{and} \quad\mathbf{b_2} = \begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}. \] The angle between them is given by: \[\cos\theta = \frac{1(2) + 2(-1) + 2(2)}{\sqrt{1^2 + 2^2 + 2^2} \cdot\sqrt{2^2 + (-1)^2 + 2^2}} = \frac{4}{3\sqrt{5}} \approx 0.5976 \]\[\theta\approx 53.3^\circ. \]
Applications to Kinematics
The vector equation of a line has direct applications in kinematics, the branch of physics dealing with motion.
Interpreting Components
In a kinematic context:
- $\mathbf{a}$ represents the initial position of an object
- $\mathbf{b}$ represents the velocity vector
- $\lambda$ represents time
The equation $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$ then describes the position $\mathbf{r}$ of an object at time $\lambda$, given its initial position $\mathbf{a}$ and velocity $\mathbf{b}$.
NoteThe magnitude of the velocity vector, $|\mathbf{b}|$, represents the speed of the object.
ExampleAn object starts at position \[\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} \] and moves with a velocity of \[\begin{bmatrix} 4 \\ -2 \\ 1 \end{bmatrix} \] units per second. Its position after \( t \) seconds is given by: \[\mathbf{r} = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} + t \begin{bmatrix} 4 \\ -2 \\ 1 \end{bmatrix} \] The speed of the object is given by: \[ |\mathbf{b}| = \sqrt{4^2 + (-2)^2 + 1^2} = \sqrt{21} \approx 4.58 \] units per second.
TipWhen solving kinematics problems, pay attention to the units. Ensure that the time unit in $\lambda$ matches the unit used in the velocity vector $\mathbf{b}$.
