Finding the Angle Between Vectors A and B Given a-b ab

Understanding the relationship between vectors in mathematical problems can provide deep insights into their geometric properties. In this article, we explore how to find the angle between two vectors A and B when given that A - B A ? B. This problem is not only fundamental in linear algebra but also has wide-ranging applications in physics and engineering. We'll delve into the steps to solve this problem and explain the significance of the results.

Introduction

In vector algebra, the dot product of two vectors provides a powerful tool to understand the relationship between them. The dot product is defined as: A ? B |A| |B| cosθ, where θ is the angle between the vectors. Given the condition A - B A ? B, our goal is to determine the angle θ between the vectors A and B.

The Problem

We start with the given equation: A - B A ? B. Our first step is to manipulate this equation to reveal the relationship between A and B.

Manipulating the Equation

Starting with the equation: A - B A ? B Square both sides: (A - B)2 (A ? B)2 Expand the left side: A2 - 2A ? B B2 (A ? B)2 Recognize that (A ? B)2 (A ? B)(A ? B), so: A2 - 2A ? B B2 (A ? B)(A ? B) Expand the right side using the dot product property: A2 - 2A ? B B2 (A2 ? B2 ? (A ? B)) Notice that A2 and B2 can be canceled from both sides: -2A ? B -2(A ? B) Thus, we simplify to: 4A ? B 0 This implies that: A ? B 0

Interpreting the Result

The dot product of two vectors is zero, A ? B 0, when the vectors are orthogonal. In geometry, orthogonality means the vectors form a 90° angle with each other. Therefore, we can conclude that the angle θ between the vectors A and B is 90°.

Conclusion

To summarize, given the condition A - B A ? B, we have shown that the angle between the vectors A and B is 90°. This result can be written as: θ 90°. Understanding this property is crucial for many applications in physics and engineering, such as analyzing the forces between objects or understanding the geometry of vector spaces.

Key Takeaways

A - B A ? B implies that the vectors A and B are orthogonal. The angle between A and B is 90°. The dot product of orthogonal vectors is zero, A ? B 0.

This article has shown the step-by-step process of solving the problem and the importance of the result in vector analysis. For a more in-depth understanding and further applications, refer to advanced resources on vector algebra and linear algebra.