Find Length Of A Vector After Recursive Addition At The Same Angle

Introduction

In geometry and vector mathematics, understanding the behavior of vectors under repeated addition is crucial for solving various problems. When a vector is added to another vector at the same angle multiple times, it's essential to determine the resulting vector's magnitude. In this article, we will explore the closed-form way to represent the magnitude of a 2D vector after another vector has been added to it at an angle of $\theta$ a total of $n$ times.

Understanding the Problem

Given a 2D vector $\vec{r}$ and another vector $\vec{a}$, we want to find the magnitude of the resulting vector after $\vec{a}$ has been added to $\vec{r}$ at an angle of $\theta$ a total of $n$ times. The vector $\vec{a}$ has a magnitude of $a$ and an angle of $\theta$ with respect to the x-axis.

Mathematical Representation

Let's represent the initial vector $\vec{r}$ as $(x, y)$ and the vector $\vec{a}$ as $(a \cos \theta, a \sin \theta)$. When $\vec{a}$ is added to $\vec{r}$ at an angle of $\theta$, the resulting vector can be represented as:

$$\vec{r}' = (x + a \cos \theta, y + a \sin \theta)$$

Recursive Addition

When $\vec{a}$ is added to $\vec{r}'$ at the same angle $\theta$, the resulting vector can be represented as:

$$\vec{r}'' = (x + 2a \cos \theta, y + 2a \sin \theta)$$

This process can be repeated $n$ times, resulting in the following vector:

$$\vec{r}^{(n)} = (x + na \cos \theta, y + na \sin \theta)$$

Magnitude of the Resulting Vector

The magnitude of the resulting vector $\vec{r}^{(n)}$ can be calculated using the Pythagorean theorem:

$$|\vec{r}^{(n)}| = \sqrt{(x + na \cos \theta)^2 + (y + na \sin \theta)^2}$$

Closed-Form Representation

To find a closed-form representation of the magnitude of the resulting vector, we can use the following trigonometric identity:

$$\cos \theta + \cos \phi = 2 \cos \left(\frac{\theta + \phi}{2}\right) \cos \left(\frac{\theta - \phi}{2}\right)$$

Applying this identity to the magnitude of the resulting vector, we get:

$$|\vec{r}^{(n)}| = \sqrt{(x + na \cos \theta)^2 + (y + na \sin \theta)^2}$$

$$= \sqrt{x^2 + y^2 + 2na(x \cos \theta + y \sin \theta) + n^2a^2}$$

$$= \sqrt{x^2 + y^2 + 2na(x \cos \theta + y \sin \theta) + n^2a^2 \cos^2 \theta + n^2a^2 \sin^2 \theta}$$

$$= \sqrt{x^2 + y^2 + 2na(x \cos \theta + y \sin \theta) + n^2a^2(\cos^2 \theta + \sin^2 \theta)}$$

$$= \sqrt{x^2 + y^2 + 2na(x \cos \theta + y \sin \theta) + n^2a^2}$$

Simplifying the Expression

Using the trigonometric identity $\cos^2 \theta + \sin^2 \theta = 1$, we can simplify the expression:

$$|\vec{r}^{(n)}| = \sqrt{x^2 + y^2 + 2na(x \cos \theta + y \sin \theta) + n^2a^2}$$

$$= \sqrt{(x + na \cos \theta)^2 + (y + na \sin \theta)^2}$$

Final Result

The magnitude of the resulting vector $\vec{r}^{(n)}$ after $\vec{a}$ has been added to it at an angle of $\theta$ a total of $n$ times is given by:

$$|\vec{r}^{(n)}| = \sqrt{(x + na \cos \theta)^2 + (y + na \sin \theta)^2}$$

$$= \sqrt{x^2 + y^2 + 2na(x \cos \theta + y \sin \theta) + n^2a^2}$$

This is the closed-form representation of the magnitude of the resulting vector.

Conclusion

In this article, we have derived a closed-form representation of the magnitude of a 2D vector after another vector has been added to it at an angle of $\theta$ a total of $n$ times. The resulting vector's magnitude can be calculated using the Pythagorean theorem and trigonometric identities. This representation can be used to solve various problems in geometry and vector mathematics.

Future Work

This work can be extended to higher-dimensional vectors and more complex addition scenarios. Additionally, the application of this work to real-world problems, such as computer graphics and physics simulations, can be explored.

References

• [1] "Vector Addition and Subtraction" by Khan Academy
• [2] "Trigonometry" by MIT OpenCourseWare
• [3] "Geometry and Trigonometry" by Wolfram MathWorld

Glossary

• Vector: A mathematical object that has both magnitude and direction.
• Magnitude: The length or size of a vector.
• Angle: The measure of the rotation of a vector from the x-axis.
• Trigonometric identity: A mathematical equation that relates the trigonometric functions of an angle.
• Pythagorean theorem: A mathematical equation that relates the lengths of the sides of a right triangle.
  

Q: What is the initial vector $\vec{r}$ and how is it represented?

A: The initial vector $\vec{r}$ is a 2D vector represented as $(x, y)$, where $x$ and $y$ are the components of the vector in the x and y directions, respectively.

Q: What is the vector $\vec{a}$ and how is it represented?

A: The vector $\vec{a}$ is a 2D vector represented as $(a \cos \theta, a \sin \theta)$, where $a$ is the magnitude of the vector and $\theta$ is the angle between the vector and the x-axis.

Q: How is the resulting vector $\vec{r}^{(n)}$ represented after $\vec{a}$ has been added to it at an angle of $\theta$ a total of $n$ times?

A: The resulting vector $\vec{r}^{(n)}$ is represented as $(x + na \cos \theta, y + na \sin \theta)$, where $x$ and $y$ are the components of the initial vector $\vec{r}$, $n$ is the number of times $\vec{a}$ has been added, and $\theta$ is the angle between $\vec{a}$ and the x-axis.

Q: How is the magnitude of the resulting vector $\vec{r}^{(n)}$ calculated?

A: The magnitude of the resulting vector $\vec{r}^{(n)}$ is calculated using the Pythagorean theorem:

$$|\vec{r}^{(n)}| = \sqrt{(x + na \cos \theta)^2 + (y + na \sin \theta)^2}$$

Q: Can the magnitude of the resulting vector $\vec{r}^{(n)}$ be represented in a closed-form?

A: Yes, the magnitude of the resulting vector $\vec{r}^{(n)}$ can be represented in a closed-form using the following expression:

$$|\vec{r}^{(n)}| = \sqrt{x^2 + y^2 + 2na(x \cos \theta + y \sin \theta) + n^2a^2}$$

Q: What is the significance of the angle $\theta$ in the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$?

A: The angle $\theta$ represents the direction of the vector $\vec{a}$ with respect to the x-axis. The magnitude of the resulting vector $\vec{r}^{(n)}$ depends on the angle $\theta$ and the number of times $\vec{a}$ has been added.

Q: Can the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ be simplified using trigonometric identities?

A: Yes, the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ can be simplified using trigonometric identities, such as the Pythagorean theorem and the trigonometric identity $\cos^2 \theta + \sin^2 \theta = 1$.

Q: What are some real-world applications of the calculation of the magnitude of the resulting vector $\vec{r}^{n)}$?

A: The calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ has applications in computer graphics, physics simulations, and engineering design.

Q: Can the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ be extended to higher-dimensional vectors?

A: Yes, the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ can be extended to higher-dimensional vectors using similar techniques.

Q: What are some limitations of the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$?

A: The calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ assumes that the vector $\vec{a}$ is added to the initial vector $\vec{r}$ at the same angle $\theta$ a total of $n$ times. If the angle $\theta$ changes or the vector $\vec{a}$ is added at different angles, the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ may not be accurate.

Q: Can the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ be used to solve problems in geometry and vector mathematics?

A: Yes, the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ can be used to solve problems in geometry and vector mathematics, such as finding the length of a vector after recursive addition at the same angle.

Q: What are some future directions for research on the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$?

A: Some future directions for research on the calculation of the magnitude of the resulting vector $\vec{r}^{(n)}$ include extending the calculation to higher-dimensional vectors, exploring applications in computer graphics and physics simulations, and developing new algorithms for calculating the magnitude of the resulting vector $\vec{r}^{(n)}$.