Problem 9 - 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. $$r=2, \alpha=60^{\circ}$$

Step-by-Step Solution

Answer

The vector in Cartesian coordinates is \( \begin{pmatrix} 1 \\ \sqrt{3} \end{pmatrix} \).

Step 1: Express Conversion Formula

To convert a vector from polar to Cartesian coordinates, we use the formulas: \( x_1 = r \cos(\alpha) \) and \( x_2 = r \sin(\alpha) \). In these formulas, \( x_1 \) and \( x_2 \) represent the Cartesian coordinates, \( r \) is the magnitude of the vector, and \( \alpha \) is the angle in degrees.

Step 2: Calculate \( x_1 \)

Substitute the given values \( r = 2 \) and \( \alpha = 60^{\circ} \) into the formula for \( x_1 \): \[ x_1 = 2 \cos(60^{\circ}) \].The cosine of \( 60^{\circ} \) is \( 0.5 \). Therefore, we calculate:\[ x_1 = 2 \times 0.5 = 1 \].

Step 3: Calculate \( x_2 \)

Substitute the given values \( r = 2 \) and \( \alpha = 60^{\circ} \) into the formula for \( x_2 \): \[ x_2 = 2 \sin(60^{\circ}) \].The sine of \( 60^{\circ} \) is \( \frac{\sqrt{3}}{2} \). Therefore, we calculate:\[ x_2 = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \].

Step 4: Combine Results

The Cartesian coordinates \( \begin{pmatrix} x_1 \ x_2 \end{pmatrix} \) of the vector are, therefore: \( \begin{pmatrix} 1 \ \sqrt{3} \end{pmatrix} \).

Key Concepts

Vector Representation

Understanding vector representation involves depicting physical quantities like force or velocity using vectors. Vectors are usually expressed as an arrow, which shows both magnitude and direction. They can be described in different coordinate systems, such as polar or Cartesian. Vectors play a crucial role in mathematics and physics, providing a way to express quantities that have both size and direction, unlike scalars which only have magnitude. When dealing with vector representation in various forms, it's important to understand how conversion works between systems.

Trigonometric Functions

Trigonometric functions are fundamental tools used in converting polar coordinates to Cartesian coordinates. They include sine (sin') and cosine (cos'), among others.

• The cosine function calculates the adjacent side over the hypotenuse in a right triangle.
• The sine function calculates the opposite side over the hypotenuse.

By using these trigonometric ratios, you can convert polar coordinates, given as a radius and an angle, into Cartesian coordinates, represented in terms of x and y. For example, with a vector magnitude of 2 and an angle of 60 degrees, cosine and sine help find the respective x and y components. The formulas \( x_1 = r \cos(\alpha) \) for the x-axis, and \( x_2 = r \sin(\alpha) \) for the y-axis, utilize these trigonometric functions to perform the conversion.

Polar Coordinates

Polar coordinates represent a point in a plane through two values: a distance from a reference point (known as the origin), and an angle from a reference direction. This coordinate system is especially useful in scenarios involving circular or rotational symmetry. The magnitude of the vector is expressed as \( r \), while the angle is denoted by \( \alpha \). Polar coordinates differ from Cartesian coordinates, which describe points using x and y axes.When tasked with converting a vector from polar to Cartesian coordinates, knowing the values of the magnitude and angle allows you to compute the Cartesian equivalents. As we've seen with an example where \( r = 2 \) and \( \alpha = 60^{\circ} \), substituting these into trigonometric conversion formulas yields the Cartesian coordinates \( x_1 = 1 \) and \( x_2 = \sqrt{3} \). Understanding these relationships makes it easier to transition between coordinate systems.