Question

Let \(n\) points be equally spaced on a circle, and let \(\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\) be the vectors from the center of the circle to these \(n\) points. Show that \(\mathbf{v}_{1}+\mathbf{v}_{2}+\cdots+\mathbf{v}_{n}=\mathbf{0}\).

Short answer

The sum of the vectors is zero because they are symmetrically distributed around the origin.

Step-by-step solution

Understanding the Vectors

Imagine a unit circle centered at the origin of a coordinate plane. If you place each point on this circle as described, the vectors are radii, forming equal angles with each other from the origin along the circle's circumference.

Determining the Angle Between Vectors

Since the points are equally spaced, the angle between any two consecutive vectors can be calculated as \( \frac{2\pi}{n} \) radians. This is because the full circle is \(2\pi\) radians and we are dividing it into \(n\) equal parts.

Representing the Vectors Mathematically

Each vector can be represented in terms of its components. The vector \( \mathbf{v}_k \) corresponding to the \(k\)th point can be given in complex form as \( e^{i\frac{2\pi k}{n}} \), where \( k = 0, 1, 2, \ldots, n-1 \).

Summing the Vectors

The task requires summing these vectors over all \(n\) points. The algebraic sum of these vectors is \( \sum_{k=0}^{n-1} e^{i \frac{2\pi k}{n}} \). This is a geometric series with the first term \( e^{0} = 1 \) and the common ratio \( r = e^{i \frac{2\pi}{n}} \).

Applying the Formula for the Sum of a Geometric Series

The sum \( S_n \) of the first \( n \) terms of a geometric series is given by the formula \( S_n = \frac{a(1-r^n)}{1-r} \) where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 1 \) and \( r = e^{i \frac{2\pi}{n}} \), thus \( S_n = \frac{1 - (e^{i\frac{2\pi}{n}})^n}{1 - e^{i\frac{2\pi}{n}}} \).

Simplifying the Result

Notice that \( (e^{i\frac{2\pi}{n}})^n = e^{i2\pi} = 1 \). Therefore, the numerator becomes \(1 - 1 = 0\). Hence, the sum of the series \( S_n \) is \(0\), implying that \( \mathbf{v}_{1} + \mathbf{v}_{2} + \ldots + \mathbf{v}_{n} = \mathbf{0} \).

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number, called the common ratio. In the context of our exercise, the elements of the series are the vectors corresponding to points on a circle, expressed in their complex number form.

Here, we have a geometric series starting with the first term 1 and a common ratio of \( e^{i \frac{2\pi}{n}} \), which represents a step around the unit circle in the complex plane.

• The formula for the sum \( S_n \) of the first \( n \) terms is \( S_n = \frac{a(1-r^n)}{1-r} \) where \( a \) is the first term and \( r \) is the common ratio.
• In our problem, \( a = 1 \) and \( r = e^{i\frac{2\pi}{n}} \), as each vector is a point expressed as a power of a complex number, moving counterclockwise around a circle.
• The fact that \( r^n = 1 \) (since a full rotation is completed) ensures this sum is zero, indicating the vectors sum to the zero vector.

Understanding how a geometric series operates allows you to see why evenly spaced vectors around a circle would sum to zero: they perfectly cancel out due to symmetry.

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to two dimensions by adding a second, imaginary dimension. They are written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).

In this exercise, complex numbers are used to represent vectors as points on the unit circle. Each vector is represented as \( e^{i\theta} \) where \( \theta \) is the angle measured in radians from the positive x-axis.

• The expression \( e^{i\theta} \) comes from Euler's formula, which links exponential functions with trigonometric functions, i.e., \( e^{i\theta} = \cos(\theta) + i \sin(\theta) \).
• Using complex numbers in this way helps diminish the complications that arise when summing vectors on a circle by leveraging the properties of exponential functions and trigonometry.

Thus, complex numbers offer a powerful method to handle circular vector summation in a two-dimensional plane.

Unit Circle

The unit circle is a circle of radius one centered at the origin of the complex plane. It is a key tool in trigonometry and complex number calculations.

For this problem, the unit circle helps us visualize and compute the vectors requested.

• Every point on the unit circle can represent a complex number of the form \( e^{i\theta} \), where \( \theta \) is the angle formed with the positive x-axis.
• The circle allows for a clear understanding of how vectors are represented as complex numbers as they are uniformly spaced around the circle.

The unit circle simplifies studying properties of trigonometric functions (sine and cosine) and is crucial for understanding rotations and transformations in the plane.

Radians

Radians are a unit of angle measurement based on the radius of a circle. One radian is the angle formed when the length of the arc is equal to the radius of the circle. This system simplifies calculations in mathematics, especially when angles are related to rotations.

In this problem, radians allow us to divide the circle into equal parts efficiently.

• The full circle is \( 2\pi \) radians, and dividing it into \( n \) equal parts gives \( \frac{2\pi}{n} \) radians between each vector.
• These radians measure angles directly in terms of the circle's circumference, making the calculation of evenly spaced points straightforward.

Using radians is not just precise; it also fits neatly with calculus and trigonometry, facilitating the connection between angles and lengths on a circle.