Problem 10 - Calculus for Biology and Medicine

Question

Vectors are given in their polar coordinate representation (length \(\boldsymbol{r}\), and angle \(\alpha\) measured counterclockwise from the positive \(x_{1}-\) axis). Find the representation of the vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) in Cartesian coordinates. $$r=3, \alpha=120^{\circ}$$

Step-by-Step Solution

Answer

The Cartesian coordinates are \([-1.5, \frac{3\sqrt{3}}{2}]\).

Step 1: Understanding Polar Coordinates

In polar coordinates, a vector is described by its length \( r \) and the angle \( \alpha \) it makes with the positive \( x_1 \)-axis. Here, the given \( r \) is 3, and \( \alpha \) is 120 degrees.

Step 2: Converting Angle to Radians

To use trigonometric functions, convert the angle from degrees to radians. The formula is \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \). Thus, \( 120^{\circ} \) becomes \( \frac{120 \times \pi}{180} = \frac{2\pi}{3} \) radians.

Step 3: Determining Cartesian Coordinates

The conversion from polar to Cartesian coordinates is given by the formulas: \( x_1 = r \cdot \cos(\alpha) \) and \( x_2 = r \cdot \sin(\alpha) \).

Step 4: Calculating \( x_1 \)

Substitute \( r = 3 \) and \( \alpha = \frac{2\pi}{3} \) into \( x_1 = r \cdot \cos(\alpha) \). \( x_1 = 3 \cdot \cos\left(\frac{2\pi}{3}\right) = 3 \cdot \left(-\frac{1}{2}\right) = -1.5 \).

Step 5: Calculating \( x_2 \)

Substitute \( r = 3 \) and \( \alpha = \frac{2\pi}{3} \) into \( x_2 = r \cdot \sin(\alpha) \). \( x_2 = 3 \cdot \sin\left(\frac{2\pi}{3}\right) = 3 \cdot \left(\frac{\sqrt{3}}{2}\right) = \frac{3\sqrt{3}}{2} \).

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative way to describe the location of a point in a plane, differing from the standard Cartesian system. Rather than dealing with strict horizontal and vertical distances, polar coordinates use:

• Radius (r): The distance from the origin to the point.
• Angle (\(\alpha\)): The counterclockwise angle measured from the positive x-axis to the line connecting the origin to the point.

This system is particularly useful in scenarios involving rotations and circular movements, as it naturally incorporates angular measurements.
Imagine you are spinning a disk on a table. The polar coordinates are perfect for stating how far from the center a point lies (radius) and at what angle relative to some fixed direction (angle).
In our exercise, the vector is defined as \(r = 3\) and \(\alpha = 120^\circ\). This means the point is 3 units away from the origin and lies 120 degrees counterclockwise from the positive x-axis which is the standard starting direction.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are essential tools in converting between polar and Cartesian coordinates. These functions relate the angles and sides of triangles, making them ideal for interpreting angle-based coordinates.

• Cosine (\(\cos\alpha\)): Provides the horizontal distance or projection from the origin based on the angle \(\alpha\).
• Sine (\(\sin\alpha\)): Offers the vertical distance or projection from the origin for the given angle.

To find Cartesian coordinates \((x_1, x_2)\) from polar coordinates, we use:

• \(x_1 = r \cdot \cos(\alpha)\)
• \(x_2 = r \cdot \sin(\alpha)\)

These equations translate the vector's polar description into the familiar x and y layout. Mathematically, by using trigonometric tables or a calculator, you can easily find how angles translate into horizontal and vertical positions.
This helps us understand that at \(\alpha = 120^{\circ}\), cosine indicates a negative value for \(x_1\), signifying a leftward position on the Cartesian plane, while sine results in a positive value for \(x_2\), showing upward positioning.

Angle Conversion

When working with angles in mathematical and engineering contexts, it's often necessary to switch between degrees and radians, the two primary units of angle measurement. While degrees are more intuitive (as circles being 360 degrees is familiar), radians often prove more natural in calculus and trigonometry due to their direct relation to the geometry of circles.

• Degrees: Commonly used in day-to-day measurements, with 360° making up a full circle.
• Radians: A more mathematically convenient unit, where the circumference of a circle corresponding to the angle subtends a length of one unit when the angle is one radian. A full circle is \(2\pi\) radians.

The conversion formula, \(\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}\), allows you to fluently transition between these units. In our example, \(120^{\circ}\) becomes \(\frac{2\pi}{3}\) radians. This change is essential because many mathematical functions (especially in scientific computing) utilize radians to ensure consistency across equations.
Think of angles in radians as directly relating to the arc length on a unit circle, making them quite practical for precise mathematical calculations.