Understanding the Angle Between Vectors A and B where AB A - B

In this article, we will explore the concept of the angle between vectors A and B when AB is equal to A - B. This will involve a step-by-step examination of vector properties, the parallelogram law of vector addition, and the cosine of the angle between vectors.

Introduction to Vector Properties and Definitions

Vectors are mathematical entities that have both magnitude and direction. When dealing with vectors, one important operation is vector subtraction, which can be understood geometrically as the addition of a vector and its negative counterpart. If we have two vectors A and B, the vector AB can be defined as A - B.

Vector Subtraction and Geometric Interpretation

Let x be the angle between vectors A and B. When AB A - B, we proceed as follows:

Square both sides of the equation to eliminate the square root: AB2 (A - B)2 AB AB A2 - 2 ABcosθ B2 2 ABcosθ -2 AB 4 ABcosθ 0 cosθ 0 [since AB ≠ 0] θ π/2

The positive solution, θ π/2, translates to a right angle (90°) between A and B.

Geometric Interpretation Using the Parallelogram Law

Another way to understand this problem is through the parallelogram law of vector addition. The two diagonals of a parallelogram formed by vectors A and B are given by A B and A - B. When A - B is considered as a single vector, it forms a right angle with A and B in the case of a rectangle (a special type of parallelogram).

Step-by-Step Derivation Using Cosine

Use the law of cosines to derive the angle:

Consider ( c A - B ) By the law of cosines: ( c^2 a^2 b^2 - 2abcos(theta) ) Substitute A - B for c: ( (A - B)^2 A^2 B^2 - 2ABcos(theta) ) Simplify the equation by canceling out the common terms: ( 2ABcos(theta) -2ABcos(theta) ) ( 4ABcos(theta) 0 ) ( cos(theta) 0 ) ( theta cos^{-1}(0) frac{pi}{2} ) or 90°

Conclusion

The angle between vectors A and B where AB A - B is 90°. This result can be derived using both algebraic manipulation and geometric reasoning involving the properties of vectors and the parallelogram law.

Related Keywords

- Vector angle

- Vector subtraction

- Parallelogram law

- Cosine of angle