Chapter 13.4, Problem 26WE

To determine

To find: The reason why the diagonal $\overrightarrow{AD}$ is the sum of $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

Explanation of Solution

Given information: $ABCD$ is a parallelogram in which two forces that are pulling an object at point $A$ are $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and the single force $\overrightarrow{AD}$ is the sum of two forces $\overrightarrow{AB}+\overrightarrow{AC}$ .

Diagram:

  

The parallelogram law of vector addition states that if two vectors are acting simultaneously at a point, then it can be written both in magnitude and direction by their adjacent sides from a point and their sum is equal to the resultant vector represented in both direction and magnitude by the diagonal of the parallelogram passing through the point.

Consider the above figure:

The vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ represents the sides $AB$ and $AC$ respectively.

According to the above law, the side $AD$ represents the resultant vector $\overrightarrow{AD}$ is:

  $\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BC}$

Thus, it proves that the diagonal $\overrightarrow{AD}$ is the sum of $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .