State parallelogram law of vector addition. Find analytically the magnitude and direction of resultant vector. Apply it to find the resultant when,
(i) Two vectors are parallel to each other
(ii) Two vectors are perpendicular to each other

Explanation

### Step-by-Step Solution

**Step 1: State the Parallelogram Law of Vector Addition**

The Parallelogram Law of Vector Addition states that if two vectors \( \vec{A} \) and \( \vec{B} \) are represented as two adjacent sides of a parallelogram, then the resultant vector \( \vec{R} \) can be represented by the diagonal of the parallelogram that starts from the same point. Mathematically, the magnitude of the resultant vector can be expressed as:

\[
|\vec{R}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta}
...