If two vectors acting simultaneously at a point can be represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that point.

Derivation of Parallelogram Law of Vector Addition using the Method of Components

To derive the parallelogram law of vector addition using the method of components, we can consider two vectors, A and B, which can be expressed in terms of their components along the x and y axes.

Step 1: Representation of Vectors A and B
Let the vector A be represented by its components as A = Ax i + Ay j, where i and j are the unit vectors along the x and y axes, respectively. Similarly, vector B can be represented as B = Bx i + By j.

Step 2: Addition of Vectors A and B
To add vectors A and B, we simply add their corresponding components. Therefore, the sum of A and B can be written as R = (Ax + Bx) i + (Ay + By) j.

Step 3: Visualization of Vector Addition using a Parallelogram
To visualize the vector addition geometrically, we can construct a parallelogram. Let the sides of the parallelogram represent vectors A and B, and the diagonal represent the resultant vector R.

Step 4: Applying the Method of Components
To apply the method of components, we draw perpendiculars from the initial point of vectors A and B to the diagonal (resultant vector R). Let the lengths of these perpendiculars be h1 and h2, respectively.

Step 5: Relationship between h1, h2, and Magnitudes of Vectors A and B
According to the properties of a parallelogram, the opposite sides are equal in length. Therefore, we have h1 = By and h2 = Ax.

Step 6: Deriving the Magnitude of Resultant Vector R
The magnitude of the resultant vector R can be found using the Pythagorean theorem, which states that the square of the hypotenuse (diagonal of the parallelogram) is equal to the sum of the squares of the other two sides. Thus, we have:

|R|^2 = h1^2 + h2^2
= (By)^2 + (Ax)^2

Taking the square root of both sides, we obtain:

|R| = √((By)^2 + (Ax)^2)

Step 7: Deriving the Direction of Resultant Vector R
To determine the direction of the resultant vector R, we can use trigonometry. The angle θ between vector A and the resultant vector R can be found using the tangent function:

tan(θ) = By / Ax

Taking the inverse tangent of both sides, we get:

θ = tan^(-1)(By / Ax)

Step 8: Final Expression of Resultant Vector R
Combining the magnitude and direction of the resultant vector R, we have:

R = √((By)^2 + (Ax)^2) * (cos(θ) i + sin(θ) j)

This expression represents the parallelogram law of vector addition, which states that the resultant vector of two vectors A and B can be obtained by drawing the sides of a parallelogram with these vectors and taking the diagonal as the resultant.