Understanding Angles Between Vectors in Geometry: A Comprehensive Guide

When dealing with vectors, one of the essential tasks is to determine the angle between them. This article explores how to calculate the angle between vectors using specific examples, such as vector A and vector B. We will also explore the calculation of the angle between vectors AB and B-A. Let's dive in and understand these concepts step by step.

Introduction to Vectors and Angles

Vectors are quantities that have both magnitude and direction. In geometry, we often need to determine the angle between vectors. This is useful in various applications, including physics, engineering, and computer science.

Problem: Finding the Angle Between Vectors A and B

In this example, we have vectors A and B, where the magnitude of A is 3 and the angle between A and B is 60 degrees (π/6). We need to find the angle between the vectors AB and B-A.

Step 1: Express Vectors in Component Form

Assuming vector A points along the x-axis, vector A can be expressed as:

A 3i

Vector B can be expressed in terms of its components as:

B (B cos(60°))i (B sin(60°))j

Given that the magnitude of B is 1, we can substitute:

B (1 * cos(60°))i (1 * sin(60°))j (1 * 0.5)i (1 * √3/2)j (0.5i) (√3/2j)

Step 2: Calculate the Vectors AB and B-A

First, calculate vector AB:

AB B - A (0.5i √3/2j) - (3i) -2.5i √3/2j

Next, calculate vector B-A:

B - A (0.5i √3/2j) - 3i -2.5i √3/2j

Notice that AB and B-A are different vectors, as B-A is the vector from point B to A, and AB is the vector from point A to B.

Step 3: Calculate the Dot Product of AB and B-A

The dot product of two vectors C and D is given by:

C · D Cx * Dx Cy * Dy

For AB and B-A:

AB · (B - A) (-2.5i √3/2j) · (-2.5i √3/2j)

AB · (B - A) (-2.5)(-2.5) (√3/2)(√3/2) 6.25 0.75 7

Step 4: Calculate the Magnitudes of AB and B-A

The magnitude of a vector C is given by:

|C| √(Cx2 Cy2)

For AB:

|AB| √((-2.5)2 (√3/2)2) √(6.25 0.75) √7

For B-A:

|B - A| √((-2.5)2 (√3/2)2) √(6.25 0.75) √7

Step 5: Use the Dot Product to Find the Angle

The angle θ between two vectors C and D can be found using the formula:

cos θ (C · D) / (|C| |D|)

Substituting the values we have:

cos θ 7 / (√7 * √7) 7 / 7 1

Therefore, θ arccos(1) 0°

Conclusion

The angle between vectors AB and B-A in this specific example is 0 degrees, as they are in the same direction. This completes our step-by-step guide to calculating the angle between vectors. If you need further assistance with any part of the calculations or have specific questions, feel free to ask!