Understanding the Alignment of Two Equal-Vagnitude Vectors for a Resultant of the Same Magnitude

Introduction

In vector mathematics, it is intriguing to explore the conditions under which two vectors of equal magnitude can produce a resultant vector of the same magnitude. This article delves into the theoretical and practical aspects of achieving this, providing insights into the specific alignment required.

Theoretical Foundation

Let us consider two vectors, A and B, both having the same magnitude. Our goal is to find the angle between these vectors such that their resultant vector also has the same magnitude as the individual vectors.

Magnitude of Vectors

Define the magnitudes of the vectors as A and B, where A B.

Resultant Vector Calculation

The resultant vector R can be calculated using the vector addition formula:

R sqrt{A^2 B^2 2AB cos(theta)}

where theta is the angle between the two vectors.

Setting the Resultant Magnitude

To ensure the resultant vector R has the same magnitude as the individual vectors, we set R A or R B. Using the above formula:

A sqrt{A^2 A^2 2A^2 cos(theta)}

Simplifying the equation:

A sqrt{2A^2(1 cos(theta))}

Squaring both sides:

A^2 2A^2(1 cos(theta))

Dividing both sides by A^2 (assuming A eq 0):

1 2(1 cos(theta))

Further simplification results in:

1 2 2 cos(theta)

Solving for theta:

2 cos(theta) -1

Hence:

cos(theta) -frac{1}{2}

Finding the Angle

The angle theta that satisfies cos(theta) -frac{1}{2} can be either 120° or 240°. However, in practical scenarios, the angle 120° is typically used for producing a resultant vector of the same magnitude.

Conclusion

Therefore, to achieve a resultant vector of the same magnitude as two equal-magnitude vectors, the two vectors must be aligned at an angle of 120° to each other.

Additional Insights

Consider two vectors A and B with magnitudes equal to each other and resultant vector C. The formula for the square of the resultant vector is:

C^2 A^2 B^2 2AB cos(theta)

If A B C, then:

1 1 1 2 cos(theta) -1 2 cos(theta) cos(theta) -frac{1}{2}

Thus, the angle is 120°. This can be generalized to any vectors by considering the direction of vector A on the x-axis and vector B at 120° with respect to the x-axis. The resultant vector will then have a magnitude of A and an angle of 60° with the x-axis.

Using these principles, one can effectively determine the alignment needed to achieve a resultant vector of the same magnitude as the individual vectors.