Chapter 5, Problem 22A

Whenever you add 3 and 4, the result is 7. This is true if the quantities being added are scalar quantities. If 3 and 4 are the magnitudes of vector quantities, when will the magnitude of their sum be 5?

To determine

The condition for the addition of magnitude of two vector quantities to obtain the given result.

Answer to Problem 22A

At $\theta =90°$ the addition of two vectors, will give the magnitude 5. This means that when the vectors are at right to each other.

Explanation of Solution

Given data:

Addition of two quantities is 7 when given quantities are 3 and 4 which is scalar.

The magnitude of two vectors are,

  $\begin{array}{l}\left|\overrightarrow{A}\right|=4\\ \left|\overrightarrow{B}\right|=3\end{array}$

Formula used:

The triangle law of vector addition is,

  

Resultant vector $\overrightarrow{C}$ is the sum of vector $\overrightarrow{A}$ and vector $\overrightarrow{B}$ as below,

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

The magnitude of resultant vector is,

  $\left|\overrightarrow{C}\right|=\sqrt{{\left|\overrightarrow{A}\right|}^2+{\left|\overrightarrow{B}\right|}^2+2\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\cos \left(\theta \right)}----------\left(1\right)$

Where, $\theta$ is the angle between extended vector $\overrightarrow{A}$ and vector $\overrightarrow{B}$

Calculation:

For the resultant vector magnitude, substitute all the values in equation (1),

  $\begin{array}{c}5=\sqrt{4^2+3^2+2\left(3\times 4\right)\cos \left(\theta \right)}\\ 25=16+9+24\cos \theta \\ 25-25=24\cos \theta \\ \cos \theta =0\\ \theta =90°\end{array}$

The angle is $90°$ for the given magnitudes. This means that the magnitude will be equal to 5 only when the vectors are perpendicular to each other.

Conclusion:

Therefore, at $\theta =90°$ the addition of two vectors, will be 5.