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

Introduction

In geometry and vector mathematics, understanding the behavior of vectors under recursive operations is crucial for solving various problems in physics, engineering, and computer science. One such problem involves finding the length of a vector after adding another vector at the same relative angle, a total of n times. In this article, we will explore the closed-form representation of the magnitude of a 2D vector after recursive vector addition at the same relative angle.

Problem Formulation

Given a 2D vector $\vec{r}$ and another vector $\vec{a}$, we want to find the length of $\vec{r}$ after adding $\vec{a}$ to it 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 denote the magnitude of the vector $\vec{r}$ after $n$ additions as $r_n$. We can represent the recursive addition of $\vec{a}$ to $\vec{r}$ as follows:

$$\vec{r}_{n+1} = \vec{r}_n + \vec{a}$$

Using the law of cosines, we can express the magnitude of $\vec{r}_{n+1}$ as:

$$r_{n+1}^2 = r_n^2 + a^2 + 2r_n a \cos(\theta)$$

Closed-Form Representation

To find a closed-form representation of the magnitude of $\vec{r}$ after $n$ additions, we can use the following recurrence relation:

$$r_{n+1}^2 = r_n^2 + a^2 + 2r_n a \cos(\theta)$$

We can rewrite this recurrence relation as:

$$r_{n+1}^2 - r_n^2 = a^2 + 2r_n a \cos(\theta)$$

Telescoping Sum

To find a closed-form representation, we can use the telescoping sum property. We can rewrite the recurrence relation as:

$$r_{n+1}^2 - r_n^2 = a^2 + 2r_n a \cos(\theta)$$

$$r_{n+2}^2 - r_{n+1}^2 = a^2 + 2r_{n+1} a \cos(\theta)$$

$$r_{n+3}^2 - r_{n+2}^2 = a^2 + 2r_{n+2} a \cos(\theta)$$

...

$$r_{n+k}^2 - r_{n+k-1}^2 = a^2 + 2r_{n+k-1} a \cos(\theta)$$

Summing the Recurrence Relation

We can sum the recurrence relation from $n+1$ to $n+k$:

$$\sum_{i=n+1}^{n+k} (r_i^2 - r_{i-1}^2) = \sum_{i=n+1}^{n+k} (a^2 + 2r_{i-1} a \cos(\theta))$$

Evaluating the Sum

We can evaluate the sum as follows:

$$\sum_{i=n+1}^{n+k} (r_i^2 - r_{i-1}^2) = r_{n}^2 - r_n^2$$

$$\sum_{i=n+1}^{n+k} (a^2 + 2r_{i-1} a \cos(\theta)) = ka^2 + 2a \cos(\theta) \sum_{i=n+1}^{n+k} r_{i-1}$$

Simplifying the Expression

We can simplify the expression as follows:

$$r_{n+k}^2 - r_n^2 = ka^2 + 2a \cos(\theta) \sum_{i=n+1}^{n+k} r_{i-1}$$

Closed-Form Representation

We can rewrite the expression as:

$$r_{n+k}^2 = r_n^2 + ka^2 + 2a \cos(\theta) \sum_{i=n+1}^{n+k} r_{i-1}$$

Geometric Series

We can recognize the sum as a geometric series:

$$\sum_{i=n+1}^{n+k} r_{i-1} = r_n + r_{n+1} + ... + r_{n+k-1}$$

Evaluating the Geometric Series

We can evaluate the geometric series as follows:

$$\sum_{i=n+1}^{n+k} r_{i-1} = r_n \frac{1 - (\frac{a}{r_n} \cos(\theta))^k}{1 - \frac{a}{r_n} \cos(\theta)}$$

Closed-Form Representation

We can rewrite the expression as:

$$r_{n+k}^2 = r_n^2 + ka^2 + 2a \cos(\theta) r_n \frac{1 - (\frac{a}{r_n} \cos(\theta))^k}{1 - \frac{a}{r_n} \cos(\theta)}$$

Simplifying the Expression

We can simplify the expression as follows:

$$r_{n+k}^2 = r_n^2 (1 + \frac{ka^2}{r_n^2} + 2 \frac{a}{r_n} \cos(\theta) \frac{1 - (\frac{a}{r_n} \cos(\theta))^k}{1 - \frac{a}{r_n} \cos(\theta)})$$

Closed-Form Representation

We can rewrite the expression as:

$$r_{n+k}^2 = r_n^2 (1 + \frac{ka^2}{r_n^2} + 2 \frac{a}{r_n} \cos(\theta) \frac{1 - (\frac{a}{r_n} \cos(\theta))^k}{1 - \frac{a}{r_n} \cos(\theta)})$$

Final Expression

We can rewrite the expression as:

$$r_{n+k}^2 = r_n^2 (1 + \frac{ka^2}{r_n^2} + 2 \frac{a}{r_n} \cos(\theta) \frac{1 - (\frac{a}{r_n} \cos(\theta))^k}{1 - \frac{a}{r_n} \cos(\theta)})$$

Conclusion

In this article, we have derived a closed-form representation of the magnitude of a 2D vector after recursive vector addition at the same relative angle. The expression is given by:

$$r_{n+k}^2 = r_n^2 (1 + \frac{ka^2}{r_n^2} + 2 \frac{a}{r_n} \cos(\theta) \frac{1 - (\frac{a}{r_n} \cos(\theta))^k}{1 - \frac{a}{r_n} \cos(\theta)})$$

This expression can be used to find the length of a vector after adding another vector at the same relative angle, a total of n times.

References

• [1] "Vector Addition and Subtraction" by Khan Academy
• [2] "Geometry and Trigonometry" by MIT OpenCourseWare
• [3] "Vector Calculus" by University of Michigan

Code

import math
def vector_addition(r_n, a, theta, k):
return r_n2 * (1 + k*a2/r_n**2 + 2a/r_nmath.cos(theta)(1 - (a/r_nmath.cos(theta))**k)/(1 - a/r_n*math.cos(theta)))
r_n = 10
a = 5
theta = math.pi/4
k = 10
result = vector_addition(r_n, a, theta, k)
print(result)

Future Work

• Investigate the behavior of the expression for different values of k and theta.
• Derive a closed-form representation for the case where the angle theta is not constant.
• Explore the application of this result in physics and engineering problems.
  

Introduction

In our previous article, we derived a closed-form representation of the magnitude of a 2D vector after recursive vector addition at the same relative angle. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the significance of the angle theta in the expression?

A: The angle theta represents the relative angle between the two vectors being added. It plays a crucial role in determining the magnitude of the resulting vector.

Q: How does the value of k affect the magnitude of the resulting vector?

A: The value of k represents the number of times the vector is added to itself at the same relative angle. As k increases, the magnitude of the resulting vector also increases.

Q: Can the expression be simplified for the case where the angle theta is 0 or pi?

A: Yes, the expression can be simplified for the case where the angle theta is 0 or pi. In this case, the expression reduces to:

$$r_{n+k}^2 = r_n^2 (1 + k \frac{a^2}{r_n^2})$$

Q: How does the magnitude of the initial vector r_n affect the magnitude of the resulting vector?

A: The magnitude of the initial vector r_n plays a crucial role in determining the magnitude of the resulting vector. As r_n increases, the magnitude of the resulting vector also increases.

Q: Can the expression be generalized for the case where the vectors are 3D?

A: Yes, the expression can be generalized for the case where the vectors are 3D. However, the expression will be more complex and will involve the dot product of the two vectors.

Q: What are some real-world applications of this result?

A: This result has several real-world applications in physics, engineering, and computer science. Some examples include:

• Modeling the motion of objects in a gravitational field
• Analyzing the behavior of electrical circuits
• Optimizing the performance of algorithms in computer science

Q: How can this result be used to solve problems in physics and engineering?

A: This result can be used to solve problems in physics and engineering by modeling the behavior of objects and systems in terms of vector addition and subtraction. For example, it can be used to analyze the motion of objects in a gravitational field, or to optimize the performance of electrical circuits.

Q: What are some limitations of this result?

A: This result has several limitations, including:

• It assumes that the vectors are 2D and that the angle theta is constant.
• It does not account for the effects of friction and other external forces.
• It is only applicable to a limited range of values for k and theta.

Q: How can this result be extended to more complex systems?

A: This result can be extended to more complex systems by using techniques such as:

• Using higher-dimensional vectors
• Introducing external forces and friction
• Using numerical methods to solve the resulting equations

Conclusion

In this article, we have answered some frequently asked questions related to the closed-form representation of the magnitude of a 2D vector after recursive vector addition at the same relative angle. We hope that this article has provided a useful resource for students and researchers in physics, engineering, and computer science.

References

• [1] "Vector Addition and Subtraction" by Khan Academy
• [2] "Geometry and Trigonometry" by MIT OpenCourseWare
• [3] "Vector Calculus" by University of Michigan

Code

import math
def vector_addition(r_n, a, theta, k):
return r_n2 * (1 + k*a2/r_n**2 + 2a/r_nmath.cos(theta)(1 - (a/r_nmath.cos(theta))**k)/(1 - a/r_n*math.cos(theta)))

r_n = 10
a = 5
theta = math.pi/4
k = 10
result = vector_addition(r_n, a, theta, k)
print(result)

Future Work

• Investigate the behavior of the expression for different values of k and theta.
• Derive a closed-form representation for the case where the angle theta is not constant.
• Explore the application of this result in physics and engineering problems.