Parallelogram Law of Vector Addition
The parallelogram law of vector addition states that if two vectors are represented by two adjacent sides of a parallelogram, then the diagonal of the parallelogram passing through their point of intersection represents the resultant vector. Mathematically, the resultant vector can be found by adding the two vectors using the parallelogram law.

Finding the Magnitude of Resultant Vector
To find the magnitude of the resultant vector using the parallelogram law, we can use the formula:
\[ |\textbf{R}| = \sqrt{|\textbf{A}|^2 + |\textbf{B}|^2 + 2|\textbf{A}||\textbf{B}|\cos\theta} \]
where \(\textbf{A}\) and \(\textbf{B}\) are the two given vectors, and \(\theta\) is the angle between them.

When Vectors are Parallel to Each Other
When two vectors are parallel to each other, the angle between them is 0 degrees. Therefore, the formula for the magnitude of the resultant vector simplifies to:
\[ |\textbf{R}| = \sqrt{|\textbf{A}|^2 + |\textbf{B}|^2 + 2|\textbf{A}||\textbf{B}|\cos0} = |\textbf{A}| + |\textbf{B}| \]
This means that when vectors are parallel, the magnitude of the resultant vector is equal to the sum of the magnitudes of the individual vectors.

When Vectors are Perpendicular to Each Other
When two vectors are perpendicular to each other, the angle between them is 90 degrees. In this case, the formula for the magnitude of the resultant vector simplifies to:
\[ |\textbf{R}| = \sqrt{|\textbf{A}|^2 + |\textbf{B}|^2 + 2|\textbf{A}||\textbf{B}|\cos90} = \sqrt{|\textbf{A}|^2 + |\textbf{B}|^2} \]
This means that when vectors are perpendicular, the magnitude of the resultant vector is equal to the square root of the sum of the squares of the magnitudes of the individual vectors.