Rotation of axis
Rotation of Axis
Rotation of axis is a concept in coordinate geometry where the coordinate axes are rotated by a certain angle while keeping the origin fixed. This transformation is used to simplify equations or to provide a different perspective on the geometric configuration of points, lines, and curves.
Understanding Rotation of Axis
When we rotate the axes, we essentially change the frame of reference for our coordinate system. The points in the plane still have the same positions relative to each other, but their coordinates change because we measure them against the new, rotated axes.
Formulas for Rotation of Axis
Let's consider a rotation by an angle $\theta$ in the counterclockwise direction. The original coordinates $(x, y)$ of a point P are transformed to new coordinates $(x', y')$ in the rotated system. The transformation formulas are derived from trigonometric relationships:
[ \begin{align*} x' &= x \cos \theta + y \sin \theta \ y' &= -x \sin \theta + y \cos \theta \end{align*} ]
Similarly, if we want to find the original coordinates $(x, y)$ from the rotated coordinates $(x', y')$, we use the inverse transformation:
[ \begin{align*} x &= x' \cos \theta - y' \sin \theta \ y &= x' \sin \theta + y' \cos \theta \end{align*} ]
Important Points to Remember
- Rotation does not change the distance between points or the shape of figures; it is an isometric transformation.
- The rotation angle $\theta$ is positive for counterclockwise rotation and negative for clockwise rotation.
- The formulas for rotation are derived from the basic trigonometric identities.
Differences Before and After Rotation
| Aspect | Before Rotation (Original Axes) | After Rotation (Rotated Axes) |
|---|---|---|
| Coordinate Axes | X-axis and Y-axis | X'-axis and Y'-axis |
| Coordinates of a Point | $(x, y)$ | $(x', y')$ |
| Orientation | Standard orientation | Rotated by angle $\theta$ |
| Equation of a Line | May have both x and y terms | Can be simplified |
| Distance between Points | Remains the same | Remains the same |
| Angle Measurement | Measured from the positive X-axis | Measured from the X'-axis |
Examples
Example 1: Rotation of a Point
Suppose we have a point P with coordinates $(2, 3)$, and we rotate our axes by $45^\circ$ counterclockwise. To find the new coordinates $(x', y')$ of P, we use the transformation formulas:
[ \begin{align*} x' &= 2 \cos 45^\circ + 3 \sin 45^\circ \ y' &= -2 \sin 45^\circ + 3 \cos 45^\circ \end{align*} ]
Since $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$, we get:
[ \begin{align*} x' &= 2 \cdot \frac{\sqrt{2}}{2} + 3 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \ y' &= -2 \cdot \frac{\sqrt{2}}{2} + 3 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \end{align*} ]
Example 2: Simplifying the Equation of a Line
Consider the equation of a line $2x + 3y - 6 = 0$. If we rotate the axes by an angle $\theta$ such that $\tan \theta = \frac{3}{2}$, we can eliminate the y-term in the new coordinates. The new equation will only have $x'$ term.
Using the transformation formulas, we can substitute $x$ and $y$ with $x'$ and $y'$:
[ 2(x' \cos \theta - y' \sin \theta) + 3(x' \sin \theta + y' \cos \theta) - 6 = 0 ]
Simplifying, we get:
[ (2 \cos \theta + 3 \sin \theta)x' + (-2 \sin \theta + 3 \cos \theta)y' - 6 = 0 ]
To eliminate the $y'$ term, we set $-2 \sin \theta + 3 \cos \theta = 0$, which is true for $\tan \theta = \frac{3}{2}$. The simplified equation in the rotated axes will be:
[ (2 \cos \theta + 3 \sin \theta)x' - 6 = 0 ]
This example demonstrates how rotating the axes can simplify the equation of a line, making it easier to analyze and solve.
In conclusion, rotation of axis is a powerful tool in coordinate geometry that can be used to simplify equations and gain new insights into geometric configurations. Understanding the transformation formulas and their applications is essential for solving problems related to this topic.