Problem 39 - Precalculus: Mathematics for Calculus

Question

Do you expect that the distance between two points is invariant under rotation? Prove your answer by comparing the distance \(d(P, Q)\) and \(d\left(P^{\prime}, Q^{\prime}\right)\) where \(P^{\prime}\) and \(Q^{\prime}\) are the images of \(P\) and \(Q\) under a rotation of axes.

Step-by-Step Solution

Answer

Yes, distance is invariant under rotation because rotation preserves distances between points.

Step 1: Understanding the Concept of Rotation

When we rotate a plane figure about the origin, the distance from the origin to any point on the figure remains constant, but the position changes. This transformation preserves distances, i.e., the distance between any two points before and after the rotation should be the same.

Step 2: Describing Points in a Coordinate System

Let the original coordinates of the points be \(P(x_1, y_1)\) and \(Q(x_2, y_2)\). After a rotation by an angle \(\theta\), the coordinates of \(P\) and \(Q\) transform to: \(P^{\prime}(x_1^{\prime}, y_1^{\prime})\) and \(Q^{\prime}(x_2^{\prime}, y_2^{\prime})\).

Step 3: Using Rotation Formulas

If \(P(x_1, y_1)\) rotates by \(\theta\) degrees, the new coordinates \((x_1^{\prime}, y_1^{\prime})\) are given by: \(x_1^{\prime} = x_1 \cos\theta - y_1 \sin\theta\) and \(y_1^{\prime} = x_1 \sin\theta + y_1 \cos\theta\). Similarly, for \(Q(x_2, y_2)\), the coordinates \((x_2^{\prime}, y_2^{\prime})\) become \(x_2^{\prime} = x_2 \cos\theta - y_2 \sin\theta\) and \(y_2^{\prime} = x_2 \sin\theta + y_2 \cos\theta\).

Step 4: Calculating Distance Before Rotation

The distance \(d(P, Q)\) between \(P\) and \(Q\) is calculated using the distance formula: \[ d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 5: Calculating Distance After Rotation

The distance \(d(P^{\prime}, Q^{\prime})\) between the images \(P^{\prime}\) and \(Q^{\prime}\) is given by the same distance formula: \[ d(P^{\prime}, Q^{\prime}) = \sqrt{(x_2^{\prime} - x_1^{\prime})^2 + (y_2^{\prime} - y_1^{\prime})^2} \] Substituting the rotated coordinates from the earlier step.

Step 6: Proving the Invariance

When calculating \((x_2^{\prime} - x_1^{\prime})^2 + (y_2^{\prime} - y_1^{\prime})^2\) using the expressions for \(x_1^{\prime}\), \(y_1^{\prime}\), \(x_2^{\prime}\), \(y_2^{\prime}\), it simplifies to \((x_2 - x_1)^2 + (y_2 - y_1)^2\). Thus, \(d(P, Q) = d(P^{\prime}, Q^{\prime})\), proving that the distance remains invariant.

Key Concepts

Rotation of Axes

Rotation of axes is a fundamental concept in geometry and physics. It involves rotating the entire coordinate system by a specific angle, usually around the origin. For example, if you have the point \( P(x_1, y_1) \) and rotate the axes by an angle \( \theta \), the coordinates of \( P \) change. This change is described by specific formulas, leading to new coordinates \( P'(x_1', y_1') \).

• The new x-coordinate becomes \( x_1' = x_1 \cos\theta - y_1 \sin\theta \).
• The new y-coordinate becomes \( y_1' = x_1 \sin\theta + y_1 \cos\theta \).

Rotating the axes helps understand phenomena from different perspectives and proves essential in various mathematical applications, including solving complex equations or transforming complicated coordinate systems.

Distance Formula

The distance formula is used to calculate the distance between two points in a coordinate plane. Assume you're given two points, \( P(x_1, y_1) \) and \( Q(x_2, y_2) \). The formula for finding the distance \( d(P, Q) \) is:
\[ d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• This formula derives from the Pythagorean theorem.
• The expression \((x_2 - x_1)^2 + (y_2 - y_1)^2\) calculates the square of the straight-line distance between the two points.

The distance formula is instrumental in various fields, including navigation, physics, and computer graphics, offering a reliable method to determine precise distances between points.

Coordinate Transformation

Coordinate transformation deals with changing from one coordinate system to another. When you perform a rotational transformation, the coordinates of points in the system also change according to specific mathematical expressions.

• For instance, moving from original coordinates \( (x_1, y_1) \) to \( (x_1', y_1') \) involves using trigonometric functions.
• These transformations allow points to be expressed in different systems, making complex problems more manageable.

Coordinate transformation is widely used in physics, robotics, and computer graphics, allowing for an adaptable approach to visualizing and solving real-world problems.

Invariance Proof

Invariance proof demonstrates that a certain property remains unchanged under specific transformations. In this exercise, we're proving the invariance of distance under rotation of axes. To do this:

• Calculate the original distance \( d(P, Q) \).
• Perform a rotation to find the new points \( P' \) and \( Q' \).
• Use the distance formula to find \( d(P', Q') \).

Upon simplifying the expression derived from the transformed coordinates, you'll notice it mirrors the original distance calculation. Hence, \( d(P, Q) = d(P', Q') \), affirming that the distance remains constant irrespective of the rotational transformation. These proofs are crucial in confirming the reliability of mathematical transformations across various applications.