Derive an expression for the magnitude and direction of the resultant vector using parallelogram law.

To derive an expression for the magnitude and direction of the resultant vector using the parallelogram law, we first need to understand the basic principles behind this law. The parallelogram law states that if two vectors are represented as two adjacent sides of a parallelogram, the resultant vector can be represented as the diagonal of that parallelogram. Let's break this down step by step.

Understanding the Vectors

Consider two vectors, **A** and **B**, originating from the same point. The lengths of these vectors represent their magnitudes, and the angle between them is denoted as θ. To visualize this, imagine drawing vector **A** horizontally and then vector **B** at an angle θ from **A**.

Constructing the Parallelogram

Next, we complete the parallelogram by drawing lines parallel to each vector from the tip of the other vector. The diagonal of this parallelogram represents the resultant vector **R**. The magnitude and direction of **R** can be derived using trigonometric principles.

Calculating the Magnitude

The magnitude of the resultant vector **R** can be found using the law of cosines. According to this law, the magnitude of the resultant vector is given by:

|R| = √(|A|² + |B|² + 2|A||B|cos(θ))

Here, |A| and |B| are the magnitudes of vectors **A** and **B**, respectively, and θ is the angle between them. This formula arises from the geometric properties of the triangle formed by the two vectors and their resultant.

Finding the Direction

To determine the direction of the resultant vector **R**, we can use the law of sines. The angle φ, which **R** makes with vector **A**, can be calculated as follows:

sin(φ) = (|B| * sin(θ)) / |R|

From this, we can derive φ:

φ = arcsin((|B| * sin(θ)) / |R|)

Alternatively, if we want to find the angle that **R** makes with vector **B**, we can use:

sin(ψ) = (|A| * sin(θ)) / |R|

Where ψ is the angle between **R** and vector **B**. This gives us a complete understanding of the direction of the resultant vector.

Example for Clarity

Let’s say we have two vectors: |A| = 5 units, |B| = 7 units, and the angle θ = 60 degrees. Plugging these values into our magnitude formula:

|R| = √(5² + 7² + 2 * 5 * 7 * cos(60°))

Calculating this gives:

|R| = √(25 + 49 + 35) = √109 ≈ 10.44 units

Now, to find the angle φ:

sin(φ) = (7 * sin(60°)) / 10.44

Calculating sin(60°) as √3/2, we can find φ and subsequently the angle ψ as well.

Summary

In summary, the parallelogram law provides a systematic way to find the resultant vector from two given vectors. By using the law of cosines for magnitude and the law of sines for direction, we can effectively analyze vector addition in a clear and structured manner. This method is not only applicable in physics but also in various fields such as engineering and computer graphics, where vector analysis is crucial.