Problem 22 - Calculus

Question

Show that the triangle with vertices at \((0,0),\left(r_{1}, \theta_{1}\right),\) and \(\left(r_{2}, \theta_{2}\right)\) has area \(A=\frac{1}{2} r_{1} r_{2} \sin \left(\theta_{2}-\theta_{1}\right)\). Find the base and height assuming \(0 \leqslant \theta_{1} \leqslant \theta_{2} \leqslant \pi\).

Step-by-Step Solution

Answer

The triangle's area is \(\frac{1}{2} r_1 r_2 \sin(\theta_2 - \theta_1)\), with base \(r_1\) and height aligned with \(r_2\sin(\theta_2 - \theta_1)\).

Step 1: Identify Triangle Vertices

The given points are \((0,0)\), \((r_1, \theta_1)\), and \((r_2, \theta_2)\) in polar coordinates.

Step 2: Convert Polar to Cartesian

Convert the polar coordinates to Cartesian. For point \((r_1, \theta_1)\), the Cartesian coordinates are \((r_1\cos(\theta_1), r_1\sin(\theta_1))\). Similarly, for point \((r_2, \theta_2)\), the Cartesian coordinates are \((r_2\cos(\theta_2), r_2\sin(\theta_2))\).

Step 3: Compute Base of the Triangle

The base of the triangle can be the distance between the points \((r_1\cos(\theta_1), r_1\sin(\theta_1))\) and \((r_2\cos(\theta_2), r_2\sin(\theta_2))\).

Step 4: Calculate Vector Representations

The vector from \((0,0)\) to \((r_1\cos(\theta_1), r_1\sin(\theta_1))\) is \((r_1\cos(\theta_1), r_1\sin(\theta_1))\) and from \((0,0)\) to \((r_2\cos(\theta_2), r_2\sin(\theta_2))\) is \((r_2\cos(\theta_2), r_2\sin(\theta_2))\).

Step 5: Apply the Cross Product Formula

To find the area of the triangle, use the formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\), where the base is the magnitude of the cross product of the two vectors formed. The formula for the cross product in 2D using sine is: \(|\vec{A} \times \vec{B}| = r_1 r_2 \sin(\theta_2 - \theta_1)\).

Step 6: Confirm the Area Formula

The magnitude of the cross product gives us the area as \(\frac{1}{2} r_1 r_2 \sin(\theta_2 - \theta_1)\). Therefore, the area \(A\) is indeed \(\frac{1}{2} r_1 r_2 \sin(\theta_2 - \theta_1)\), confirming the given formula.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. While Cartesian coordinates use an "x" and a "y" value to describe a point's position, polar coordinates use a radius and an angle. The radius, often denoted as "r," measures the distance from the reference point, called the pole, usually located at (0,0). The angle, typically denoted as "\(\theta\)," describes the rotation from the positive x-axis.

The conversion from polar coordinates \((r,\theta)\) to Cartesian coordinates \((x,y)\) uses the following formulas:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

To understand polar coordinates more clearly, think of drawing a line from the origin to a point. The length of the line is "r," and the angle it makes with the x-axis is "\(\theta\)."

This coordinate system is particularly useful in problems involving circular motion or symmetry. In this specific exercise, understanding polar coordinates helps translate point positions into Cartesian coordinates, which is necessary for further calculations.

Cross Product in 2D

The cross product is a vector operation used in mathematics to find the product of two vectors. In 2-dimensional space, though, the cross product doesn't result in another vector but rather a scalar—a single numerical value. This scalar helps measure the area spanned by the two vectors when they are placed tail-to-tail.

For two vectors in 2D, \(\vec{A} = \langle a_1, a_2 \rangle\) and \(\vec{B} = \langle b_1, b_2 \rangle\), the cross product can be calculated as:

• \(|\vec{A} \times \vec{B}| = a_1 b_2 - a_2 b_1\)

When representing problems geometrically, this value is equivalent to the area of the parallelogram formed by the vectors \(\vec{A}\) and \(\vec{B}\).

In our specific exercise of finding the triangle's area using polar coordinates, we exploit this by using the formula \(|\vec{A} \times \vec{B}| = r_1 r_2 \sin(\theta_2 - \theta_1)\). The idea here is that the cross product measures how much of one vector is "outside" the other, thus providing the distance (or height) necessary for area calculation.

Vector Representation

Vectors are essential mathematical objects that have both magnitude and direction. In the context of geometric problems, they help represent segments or movements on the plane. Each vector can be described in Cartesian coordinates using its x and y components.

In our exercise, the points of interest are initially in polar form, like (\(r, \theta\)). To find a triangle's area formed by these points, converting these points into vector representations is crucial. This is achieved by understanding a vector as starting from the origin \((0, 0)\) and reaching the point given in Cartesian coordinates.

For example, if you have the point \((r_1, \theta_1)\), its vector representation in Cartesian form would be \((r_1 \cos(\theta_1), r_1 \sin(\theta_1))\). This vector expresses the position in terms of horizontal and vertical displacements.

Understanding vector representation allows us to perform operations like the cross product, enabling our computation of areas in geometrical figures. In our case, expressing vectors originating from polar coordinates allows us to geometrically interpret the problem and verify the formula for the triangle's area.