Determining the Angle Between Two Vectors for Equal Magnitude Resultants

In the realm of mathematics and physics, understanding the relationship between vectors, their angles, and the magnitudes of their resultants is crucial. This article explores the specific scenario where the magnitude of the resultant of the difference between two vectors is equal to the magnitude of either vector. We will delve into the underlying principles and provide a clear, step-by-step solution.

The Problem: Equal Magnitude Resultant of Vectors

Consider two vectors, a and b, which represent physical quantities with magnitude and direction. The resultant of vector subtraction, a - b, is given to have the same magnitude as either a or b. The challenge is to determine the angle between these vectors, denoted by θ, that satisfies this condition.

Mathematical Formulation and Solution

Let the two vectors be vec{A} and vec{B}. We need to find the angle θ_{AB} such that the resultant of the difference between these vectors, vec{A} - vec{B}, has a magnitude equal to the magnitude of either vector.

The magnitude of the resultant vector R is given by:

R sqrt{A^2 B^2 - 2AB cos θ}

Given that vec{A} - vec{B} vec{A} vec{B}",

we can write:

A^2 A^2 B^2 - 2AB cos θ 0 B^2 - 2AB cos θ 2AB cos θ B^2 cos θ frac{B^2}{2AB} cos θ frac{B}{2A}

Solving for θ, we get:

θ 120°

Since cos 120° -frac{1}{2}, the angle between the vectors is 120°.

Visualizing the Solution

Let's visualize this using a graphical approach. When drawing the vectors vec{A} and vec{B} side by side (tail to tail), the resultant of vec{A} - vec{B} will connect the tail of vec{A} to the tip of vec{B}. In this configuration, the angle between vec{A} and vec{A} - vec{B} is 60°, which confirms our solution.

Additional Insight: Vector Subtraction and Resultants

Another way to understand this problem is to consider the triangle formed by the vectors vec{A} and vec{B}. If the magnitude of vec{A} - vec{B} is equal to the magnitude of either vector, it forms an equilateral triangle, with each side having the same length. In an equilateral triangle, the angle between any two sides is always 60°. Thus, the angle θ_{AB} is 60°.

Therefore, the angle between two vectors vec{A} and vec{B} such that the magnitude of their resultant difference is equal to the magnitude of either vector is 60°.

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References and Further Reading

For further study, consider exploring the following concepts:

Vectors on MathIsFun Understanding Vectors and Resultants in Physics