Chapter 5.1, Problem 17SSC

To determine

Whether the resultant displacement of two unequal displacements be zero.

Whether the resultant displacement of three unequal displacements be zero.

Explanation of Solution

Introduction Let the displacements be:

  ${\overrightarrow{d}}_1$ , ${\overrightarrow{d}}_2$ and ${\overrightarrow{d}}_3$

They have unequal magnitude.

Resultant displacement can be obtained by:

  $\overrightarrow{d}={\overrightarrow{d}}_1+{\overrightarrow{d}}_2+{\overrightarrow{d}}_3$

  $d=\sqrt{d_1{}^2+d_2{}^2+d_3{}^2}$

In case of two displacement vectors, the resultant of two unequal magnitude vectors can never be zero. Even if the displacements are in opposite direction, because the magnitudes are not equal, the resultant would be non-zero.

  $\overrightarrow{d}={\overrightarrow{d}}_1+{\overrightarrow{d}}_2$

  

In case of three displacement vectors, the resultant can be zero if the resultant of two displacements is equal in magnitude but opposite in direction to the third displacement vector.

If $|{\overrightarrow{d}}_1+{\overrightarrow{d}}_2|=\left|{\overrightarrow{d}}_3\right|$

  $\overrightarrow{d}={\overrightarrow{d}}_1+{\overrightarrow{d}}_2-{\overrightarrow{d}}_3=0$