Understanding Vectors: When Vector Addition Equals Vector Subtraction

Understanding Vectors: When Vector Addition Equals Vector Subtraction

Introduction to Vectors

Vectors are an essential concept in various fields, including physics, engineering, and mathematics. They represent quantities that have both magnitude and direction. The relationship between the angles at which vectors are placed can change how these vectors combine, whether through addition or subtraction. This article delves into the special condition under which the magnitude of the addition of two vectors equals the magnitude of their subtraction.

The Condition for Vector Addition Equaling Vector Subtraction

For the magnitudes of the sum and difference of two vectors to be equal, a specific geometric condition must be satisfied. Specifically, this condition occurs when the angle between the two vectors is 90 degrees, or equivalently, pi/2 radians.

Geometric Interpretation

When two vectors form a 90-degree angle, their resultant vector addition and subtraction can be understood through the Pythagorean theorem. If we represent the two vectors as 'a' and 'b', and the resultant vector from their addition as 'c', then the magnitudes must satisfy the relationship:

Mathematical Proof

Consider two vectors a and b. The magnitude of their sum, c, can be calculated using the formula:

c |a b| sqrt(a^2 b^2 2abcosθ), where θ is the angle between the vectors.

The magnitude of their difference, d, is given by:

d |a - b| sqrt(a^2 b^2 - 2abcosθ).

For these magnitudes to be equal, the equations must hold:

sqrt(a^2 b^2 2abcosθ) sqrt(a^2 b^2 - 2abcosθ).

Squaring both sides, we get:

a^2 b^2 2abcosθ a^2 b^2 - 2abcosθ.

This simplifies to:

4abcosθ 0.

Since a and b are non-zero vectors, the only solution is:

cosθ 0.

Thus, θ 90 degrees or π/2 radians.

Pythagorean Theorem and Vectors

When the angle θ between two vectors is 90 degrees, the Pythagorean theorem dictate that the magnitudes will be equal. Let's revisit the key equation:

c d sqrt(a^2 b^2)

This relationship is visually evident in a right-angled triangle, where a and b form the legs and c is the hypotenuse.

Applications in Physics and Engineering

This specific geometric configuration has significant real-world applications. For example, in physics, it can be used in analyzing forces acting on an object. If two force vectors form a 90-degree angle and their magnitudes are the same, the resultant force can be easily calculated using this property.

Conclusion

In conclusion, the magnitude of the sum of two vectors will equal the magnitude of their difference when the vectors form a 90-degree angle. This special case can be derived using the Pythagorean theorem. The angle of 90 degrees ensures that the vectors are perpendicular to each other, resulting in orthogonal components that simplify the vector calculations.

Key Takeaways

The magnitude of vector addition equals the magnitude of vector subtraction when the angle between the vectors is 90 degrees (π/2 radians). The Pythagorean theorem is used to prove this condition. Vectors forming a 90-degree angle can simplify complex vector problems.

Keywords: vectors, vector addition, vector subtraction