Chapter 5, Problem 92A

To determine

The change in the resultant displacement, when the angle between the vectors is increased from $0^{\circ }$ to ${180}^{\circ }$ .

Answer to Problem 92A

The angle between two vectors increases from $0^{\circ }$ to ${180}^{\circ }$ , then their resultant displacement also increases.

Explanation of Solution

Given:

Consider two vectors $\text{P}$ and $\text{Q}$ .

Formula used:

Cosine Law of Vectors:

Show the resultant displacement between two vectors $\text{<mover accent="true"><mo>P</mo><mo>→</mo></mover>}$ and $\text{<mover accent="true"><mo>Q</mo><mo>→</mo></mover>}$ as shown below.

  

The resultant for the two vectors is given by the expression.

  $\text{R=}\sqrt{{\text{P}}^{\text{2}}{\text{+ Q}}^{\text{2}}-\text{2PQ cos}\theta }$

Calculation:

Consider two vectors $\text{<mover accent="true"><mo>P</mo><mo>→</mo></mover>}$ and $\text{<mover accent="true"><mo>Q</mo><mo>→</mo></mover>}$ .

Consider the angle between the vectors is $\theta$ .

Assume that initially the angle between the vectors $\text{P}$ and $\text{Q}$ is $0°$ and the corresponding resultant is ${\text{R}}_1$ .

  $\begin{array}{c}{\text{R}}_1{}^{\text{2}}{\text{= P}}^{\text{2}}{\text{+ Q}}^{\text{2}}-\text{ 2PQ cos}\theta \\ ={\text{P}}^{\text{2}}{\text{+ Q}}^{\text{2}}-\text{ 2PQ cos}\left(0°\right)\\ ={\text{P}}^{\text{2}}{\text{+ Q}}^{\text{2}}-\text{ 2PQ}\\ \text{ = (P}-{\text{Q)}}^2\\ {\text{ R}}_1\text{= (P}-\text{Q)}\end{array}$

Consider that the angle between the vectors are increased from $0°$ to

  $180°$ .

Assume that when the angle between the vectors $\text{P}$ and $\text{Q}$ is $180°$ , the corresponding resultant is ${\text{R}}_2$ .

  $\begin{array}{c}{\text{R}}_2{}^{\text{2}}{\text{= P}}^{\text{2}}{\text{+ Q}}^{\text{2}}-\text{2PQ cos}\theta \\ ={\text{P}}^{\text{2}}{\text{+ Q}}^{\text{2}}-\text{2PQ cos}\left(180°\right)\\ ={\text{P}}^{\text{2}}{\text{+ Q}}^{\text{2}}\text{+}\text{2PQ}\\ \text{ = (P}+{\text{Q)}}^2\\ {\text{ R}}_2\text{= (P}+\text{Q)}\end{array}$

Compare ${\text{R}}_2$ and ${\text{R}}_1$ .

  ${\text{R}}_2>{\text{R}}_1$

Hence, when the angle between two vectors increases from $0^{\circ }$ to ${180}^{\circ }$ , then their resultant displacement also increases.

Conclusion:

Thus, when the angle between two vectors increases from $0^{\circ }$ to ${180}^{\circ }$ , then their resultant displacement also increases.