under what condition the direction of the sum and differnce of two vectors are same.prove it through paraallelogram law of vector addiion

To determine when the direction of the sum and difference of two vectors is the same, we can analyze the situation using the parallelogram law of vector addition. This law states that if two vectors are represented as two adjacent sides of a parallelogram, the diagonal of that parallelogram represents their sum. The difference of the two vectors can also be visualized in a similar manner. Let's break this down step by step.

Understanding Vector Addition and Subtraction

Consider two vectors, **A** and **B**. The sum of these vectors, denoted as **A + B**, results in a vector that points in a direction that is influenced by both **A** and **B**. Conversely, the difference, **A - B**, can be thought of as adding **A** to the negative of **B** (or **A + (-B)**). The direction of these vectors can be analyzed geometrically.

Visualizing with the Parallelogram Law

According to the parallelogram law, if you draw vectors **A** and **B** originating from the same point, you can form a parallelogram. The diagonal from that point to the opposite corner represents the vector sum **A + B**. The diagonal representing the difference **A - B** can be found by taking **B** in the opposite direction and forming another parallelogram.

Condition for Same Direction

The directions of **A + B** and **A - B** will be the same under specific conditions. This occurs when the angle between the two vectors is either 0 degrees (they point in the same direction) or 180 degrees (they point in opposite directions). Let's analyze these cases:

• Case 1: Vectors in the Same Direction

  If **A** and **B** are in the same direction, the sum **A + B** will be a vector that is longer than either **A** or **B**, while the difference **A - B** will point in the same direction as **A** but will have a magnitude that is the difference of their lengths.

• Case 2: Vectors in Opposite Directions

  If **A** and **B** are in opposite directions, the sum **A + B** will yield a vector that points in the direction of the larger vector, while the difference **A - B** will point in the direction of the larger vector as well, but with a magnitude that reflects the difference.

Mathematical Proof

Let’s denote the magnitudes of **A** and **B** as |A| and |B|, and the angle between them as θ. The sum and difference can be expressed in terms of their magnitudes and the cosine of the angle between them:

• Magnitude of the sum:

  |A + B| = √(|A|² + |B|² + 2|A||B|cos(θ))

• Magnitude of the difference:

  |A - B| = √(|A|² + |B|² - 2|A||B|cos(θ))

For the directions of **A + B** and **A - B** to be the same, the angle θ must be 0 or 180 degrees. In these cases, the cosine values yield:

• For θ = 0°:

  cos(0) = 1, leading to both vectors pointing in the same direction.

• For θ = 180°:

  cos(180) = -1, leading to both vectors pointing in opposite directions.

Conclusion

In summary, the directions of the sum and difference of two vectors are the same when the vectors are either parallel or anti-parallel. This can be visually and mathematically confirmed using the parallelogram law of vector addition. Understanding these relationships helps in various applications, from physics to engineering, where vector analysis is crucial.