Problem 11 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The Cartesian coordinates of the vector are \([-\frac{1}{2}, \frac{\sqrt{3}}{2}]\).

Step 1: Understand Polar and Cartesian Coordinates

In polar coordinates, a vector is represented by its magnitude \( r \) and angle \( \alpha \). To convert this to Cartesian coordinates, we need to find the equivalent \( x_1 \) and \( x_2 \) components on an \( x-y \) plane.

Step 2: Use the Conversion Formulas

The formulas to convert from polar to Cartesian coordinates are: \( x_1 = r \cdot \cos(\alpha) \) and \( x_2 = r \cdot \sin(\alpha) \). Here, \( r = 1 \) and \( \alpha = 120^{\circ} \).

Step 3: Convert Angle to Radians

Because trigonometric functions typically use radians, convert \( \alpha = 120^{\circ} \) to radians. Use the conversion factor \( \frac{\pi}{180} \). Thus, \( \alpha = 120 \times \frac{\pi}{180} = \frac{2\pi}{3} \; \text{radians} \).

Step 4: Calculate \( x_1 \)

Use the formula \( x_1 = r \cdot \cos(\alpha) \). Plugging in the values, \( x_1 = 1 \cdot \cos(\frac{2\pi}{3}) \). Since \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \), thus \( x_1 = -\frac{1}{2} \).

Step 5: Calculate \( x_2 \)

Use the formula \( x_2 = r \cdot \sin(\alpha) \). Plugging in the values, \( x_2 = 1 \cdot \sin(\frac{2\pi}{3}) \). Since \( \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} \), thus \( x_2 = \frac{\sqrt{3}}{2} \).

Step 6: Write the Cartesian Coordinates

The Cartesian representation of the vector is \( \left[\begin{array}{c} x_1 \ x_2 \end{array}\right] = \left[\begin{array}{c} -\frac{1}{2} \ \frac{\sqrt{3}}{2} \end{array}\right] \).

Key Concepts

Rectangular Coordinates

Understanding rectangular coordinates is crucial when dealing with vectors. This system represents points on a plane using two numbers. These numbers are known as coordinates, typically denoted as \(x_1\) and \(x_2\). Rectangular coordinates are often used in mathematics due to their straightforward and universally understood nature.

• First coordinate (\(x_1\)): Represents the horizontal position on the x-axis.
• Second coordinate (\(x_2\)): Represents the vertical position on the y-axis.

Rather than thinking about direction and magnitude as in polar coordinates, rectangular coordinates break down a vector into its component parts. This method focuses on how far along and how far up an object is, from a starting point at the origin.

Trigonometric Functions

Trigonometric functions like cosine (\(\cos\)) and sine (\(\sin\)) play a significant role in converting polar coordinates to rectangular ones. They relate an angle to the ratios of the sides of a right triangle. This triangle is conceptualized by dropping a perpendicular from the position marked by the angle to the axis. These functions allow us to express vectors in different forms based on context and requirement.

• \(\cos\): Determines the horizontal component (\(x_1\)) relative to the angle.
• \(\sin\): Determines the vertical component (\(x_2\)) relative to the angle.

In a right triangle formed by the vector:- \(\cos(\alpha)\) gives the adjacent side over hypotenuse,- \(\sin(\alpha)\) gives the opposite side over hypotenuse,Thus, they are integral when solving problems of angle conversion and breaking down polar vectors.

Angle Conversion

Angles are generally given in degrees, but trigonometric functions in mathematics often use radians. Converting between these two units is essential for proper calculations. An easy way to remember the conversion is that \(360^\circ = 2\pi\) radians. Therefore, to convert degrees into radians, use the angle in degrees, multiply it by \(\frac{\pi}{180}\). For example, \(120^\circ\) as radians would be: \[120 \times \frac{\pi}{180} = \frac{2\pi}{3} \\]Converting angles is crucial when working with trigonometric functions since they rely on radian measure for calculation. This ensures that when you insert an angle into a trigonometric function, the output relates correctly to the desired measurements on a Cartesian plane.

Vector Representation

A vector in the context of coordinate systems shows direction and magnitude. It can be represented in two primary forms: polar and Cartesian.

• **Polar Representation**: Consists of the magnitude (length) \(r\) and the angle \(\alpha\) from a reference direction.
• **Cartesian Representation**: Breaks down these into \(x_1\) and \(x_2\) components that sit on perpendicular axes.

When converting a polar vector into Cartesian form, the idea is to find out how far the vector stretches horizontally and vertically away from the origin. Using trigonometry, \(x_1\) and \(x_2\) are calculated using the cosine and sine of the angle, providing a clear picture of where that vector lands on the grid. This decomposition into coordinates allows for easier manipulation and combination of vectors in further math or physics problems.