Problem 11 - 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. $$r=1, \alpha=180^{\circ}$$

Step-by-Step Solution

Answer

The Cartesian coordinates are \([-1, 0]\).

Step 1: Understand Polar Coordinates

A vector in polar coordinates is represented by its length, \( r \), and an angle, \( \alpha \), from the positive \( x_1 \)-axis. In this case, \( r = 1 \) and \( \alpha = 180^\circ \).

Step 2: Conversion Formulas

To convert polar coordinates to Cartesian coordinates, use the formulas: \( x_1 = r \cos(\alpha) \) and \( x_2 = r \sin(\alpha) \).

Step 3: Calculate \( x_1 \)

Since \( r = 1 \) and \( \alpha = 180^\circ \), calculate \( x_1 \) using \( x_1 = 1 \cdot \cos(180^\circ) \). The cosine of \( 180^\circ \) is \(-1\). Therefore, \( x_1 = -1 \).

Step 4: Calculate \( x_2 \)

Using the same values, calculate \( x_2 = 1 \cdot \sin(180^\circ) \). The sine of \( 180^\circ \) is \(0\). So, \( x_2 = 0 \).

Step 5: Result in Cartesian Coordinates

The vector in Cartesian coordinates is \( \begin{bmatrix} -1 \ 0 \end{bmatrix} \).

Key Concepts

Understanding Polar Coordinates

Polar coordinates provide a way to represent vectors using a distance and a direction. Imagine you have a point in a plane. With polar coordinates, we determine this point by knowing how far it is from a reference point (usually called the origin) and in what direction it is from a reference direction (usually the positive x-axis). For example, in our exercise, the vector is located a distance of 1 unit from the origin, and the direction is given by an angle of \(180^\circ\). This angle is measured counterclockwise from the positive x-axis, which is a common convention. When speaking of polar coordinates:

• \(r\) is the radius or the distance from the origin to the point.
• \(\alpha\) is the angle representing the direction of the vector.

Thus, polar coordinates are particularly useful in scenarios where problems exhibit circular symmetry or when dealing with circular motions. Breaking up a problem with polar coordinates allows you to easily describe both position and movement especially in radial systems.

Cartesian Coordinates Explained

Cartesian coordinates offer a different way to define points in a plane. In contrast to polar coordinates, which use radius and angle, Cartesian coordinates express the position using two perpendicular lines: the horizontal \(x_1\)-axis and the vertical \(x_2\)-axis. Each point is defined by how far it resides from these two axes.
To convert from polar to Cartesian coordinates, you utilize trigonometric functions:

• For the \(x_1\) coordinate, use the formula \(x_1 = r \cos(\alpha)\).
• For the \(x_2\) coordinate, use the formula \(x_2 = r \sin(\alpha)\).

Using our example, when \(r = 1\) and \(\alpha = 180^\circ\), we apply these formulas:

• \(x_1 = 1 \cdot \cos(180^\circ) = -1\).
• \(x_2 = 1 \cdot \sin(180^\circ) = 0\).

Thus, the Cartesian coordinates for this vector are \(\begin{bmatrix} -1 \ 0 \end{bmatrix}\). These coordinates make it possible to visualize and compute with vectors in a more straightforward manner when dealing with problems aligned with a rectangular grid, such as most coordinate system problems.

Vector Representation in Different Systems

Vector representation is a fundamental concept in mathematics and physics. Whether using polar or Cartesian coordinates, vectors maintain a consistent way to express magnitude and direction, essential for various applications.
**Polar vs. Cartesian Representation:**

• **Polar Coordinates** - Useful when dealing with rotations or circular movements. It emphasizes direction and distance from a point.
• **Cartesian Coordinates** - Ideal for standard applications involving linear calculations or when precise points need to be plotted on a grid.

Each system has its advantages based on the context of the problem you're trying to solve. Often in more complex calculations or graphical representations, changing between these representations is necessary.
**Additional Uses and Contexts:**

• In physics, polar representations are particularly advantageous for problems involving moderate to large rotations.
• Conversely, engineering often favors Cartesian coordinates, especially in structural design and alignment tasks.

Understanding how to convert between these systems allows you to decide which is more suitable for any given problem and carry out tasks effectively in varied scientific and engineering contexts.