Understanding the Vector Resultant vs. Angle Bisector

When discussing vector addition, it is often mistakenly assumed that the resultant vector always lies along the angle bisector between the original vectors. In reality, the direction of the resultant vector is determined by the specifics of the addition process, not by the angle bisector.

Vector Addition Process

The resultant vector from adding two vectors, A and B, is found by the method of vector addition. This process involves placing the tail of one vector at the head of the other and drawing a vector from the tail of the first to the head of the second. Mathematically, this resultant vector, denoted by R, is given by:

R A B

The Concept of Angle Bisector

The angle bisector of two vectors is a line that divides the angle between them into two equal parts. This concept is different from the resultant vector direction and is typically derived through normalization and scaling of the vectors.

Relationship Between Resultant Vector and Angle Bisector

The resultant vector, R, points in a direction that depends on the magnitudes and directions of vectors A and B. On the other hand, the angle bisector represents a compromise between the two angles and doesn’t generally lie in the direction of the resultant vector unless the vectors are of equal magnitude and direction.

Special Case: Vectors of Equal Magnitude

There is a special case where the resultant vector can align with the angle bisector: when vectors A and B are of equal magnitude and symmetrically positioned, such as forming an isosceles triangle. However, this scenario is not a general rule and does not apply in most cases.

Mathematical Explanation of the Special Case

To understand the mathematical basis for this special case, consider two vectors, A and B. Using the head-to-tail method, we can represent the resultant vector:

Let’s draw a perpendicular from the point where the head of B meets R on the coaxial line to A, at point Q. The angle alpha is given by:

Tan α frac{B sin θ}{A cdot B cos θ}

If A B, the equation simplifies to:

Tan α Tan (θ/2)

This implies that:

Tan α - Tan (θ/2) 0

Further simplification leads to:

Sin α cdot Cos (θ/2) - Sin (θ/2) cdot Cos α 0

Which simplifies to:

α θ/2

This confirms that the resultant vector aligns with the angle bisector when A and B have the same magnitude and are symmetrically aligned.

Conclusion

In summary, the resultant vector direction depends on the magnitudes and directions of the original vectors. The angle bisector is a different line that is a compromise between the angles, and it is only in a special case that the resultant vector aligns with the angle bisector. Understanding these differences is crucial for accurately representing and calculating vector operations.