Chapter 3, Problem 7Q

Two vectors have length $V_1=3.5\text{km}$ and $V_2=4.0\text{km}$ . What are the maximum and minimum magnitudes of their vector sum?

To determine

The maximum and minimum magnitudes of the sum of two vectors.

Answer to Problem 7Q

Solution:

The maximum is: $\left|V\right|=7.5\left[km\right]$

The minimum is: $\left|V\right|=1.5\left[km\right]$

Explanation of Solution

The maximum occurs when both vectors are exactly pointing in the same direction and the minimum when they are exactly pointing in the opposite one.

Given:

The magnitude of V₁ is 3.5 km and the magnitude of V₂ is 4.0 km.

Formula used:

To calculate the magnitude of the sum of two vectors using their magnitude we need to know the angle between the vectors:

                           $\left|V\right|=\left|\left|V_1\right|+\left|V_2\right|\cos \theta \right|$

Where $\theta$ is the difference between the angles vectors:

                           $\theta ={\theta }_1-{\theta }_2$

Where ${\theta }_1$ is the angle between V₁ and the horizontal and ${\theta }_2$ is the angle between V₂ and the horizontal. the maximum magnitude occurs when the difference is zero and the minimum when the difference is 180°. When $\theta =0°$

                   $\left|V\right|=\left|\left|V_1\right|+\left|V_2\right|\cos 0°\right|=\left|\left|V_1\right|+\left|V_2\right|\right|$

And when $\theta =180°$

                   $\left|V\right|=\left|\left|V_1\right|+\left|V_2\right|\cos 180°\right|=\left|\left|V_1\right|-\left|V_2\right|\right|$

Calculations:

The maximum is:

                   $\left|V\right|=\left|3.5\left[km\right]+4.0\left[km\right]\right|=7.5\left[km\right]$

The minimum is:

                   $\left|V\right|=\left|3.5\left[km\right]-4.0\left[km\right]\right|=1.5\left[km\right]$