Question

State and prove parallelogram law of vector addition

Answer 1

Step 1: State the law

The parallelogram of vector addition states that the sum of two vectors is the vector that represents the diagonal of a parallelogram whose adjacent sides are the addends.

Step 2: Prove the law

Let $\overrightarrow{A}$ and $\overrightarrow{B}$ be two vectors at $P$.
Let the length of $PQ$ and $PS$ be the magnitude of vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ respectively.
Draw $QR$ and $SR$ parallel and equal to $PS$ and $PQ$ respectively.
Connect $P$ and $R$ which will form the diagonal of $▱PQRS$ denoting the vector $\overrightarrow{PR}$.

We have,
$\begin{array}{c}\overrightarrow{QR}=\overrightarrow{PS}=\overrightarrow{B}\\ \overrightarrow{PQ}=\overrightarrow{SR}=\overrightarrow{A}\end{array}$

From the triangle law of vector addition,
$\overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR}$

But, $\overrightarrow{PQ}=\overrightarrow{A}$ and $\overrightarrow{QR}=\overrightarrow{B}$. Thus,

$\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{PR}$