Understanding the Angle Between Vectors A and B: A. B 1 Case

In this article, we will delve into the process of calculating the angle between vectors A and B when their dot product is given. Specifically, we will explore the case where A. B 1, and A 2, B 1. Understanding this concept is essential for anyone working with vector operations in mathematics and physics. Let's break down the steps to find the angle θ between the vectors A and B.

1. Understanding the Dot Product and Cosine Law

The dot product of two vectors A and B is defined as:

A. B |A| |B| cos θ

where |A| and |B| are the magnitudes of vectors A and B, respectively, and θ is the angle between them. This equation will be our primary tool in solving for the angle θ.

2. Calculating the Angle θ

Given the values A 2, B 1, and A. B 1, we can substitute these into the dot product equation:

A. B 1 2 * 1 * cos θ

By simplifying the equation, we get:

1 2 cos θ

Dividing both sides by 2, we obtain:

cos θ 1/2

Now, to find θ, we take the inverse cosine (arc cos) of 1/2:

θ arc cos (1/2)

Since the range of θ is 0° ≤ θ ≤ 180°, the angle θ that satisfies cos θ 1/2 within this range is:

θ 60°

3. Verification and Practical Applications

To verify our solution, let's re-calculate using another method. Given A 2, B 1, and A. B 1, we can use the following formula:

A. B A B cos θ

Substituting the given values, we have:

1 2 * 1 * cos θ

1 2 cos θ

cos θ 1/2

θ arc cos (1/2) 60°

Thus, the angle between vector A and vector B when A. B 1, A 2, and B 1 is indeed 60°.

4. Conclusion and Further Reading

Understanding the angle between vectors is crucial in various fields, such as computer graphics, physics, and engineering. The process of calculating angles using the dot product is a fundamental concept in vector analysis and should be mastered by students and professionals alike.

To deepen your understanding of vector mathematics and calculations, consider exploring related topics such as vector addition, cross product, and vector projections. Additionally, you might find it helpful to familiarize yourself with software tools like MATLAB or Python for performing vector operations more efficiently.

References

For further reading and more detailed explanations, refer to:

Math Is Fun: Vectors Khan Academy: Angle Between Vectors University of British Columbia: Vectors in Euclidean Space