Question

Suppose you add two vectors A and B. What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?

Short answer

The relative direction for maximum magnitude $\left(A+B\right)$should be in the same direction and the relative direction for minimum magnitude $\left(A-B\right)$ should be in the opposite direction.

Step-by-step solution

Identification of given data

The magnitude of first vector is A.

The magnitude of second vector is B.

The displacement is resolved along the horizontal and vertical directions; then the sum of all the horizontal components provides the horizontal component of net displacement. The sum of the vertical components provides the vertical component of net displacement.

Determination of resultant for maximum magnitude

The resultant of two vectors is given by

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

The magnitude of resultant is given by

$$\begin{gathered} {\left|\overrightarrow{R}\right|}^2={\left|\overrightarrow{A}\right|}^2+{\left|\overrightarrow{B}\right|}^2+2\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\cos \theta \\ {R}^2=A^2+B^2+2AB\cos \theta \\ R=\sqrt{A^2+B^2+2AB\cos \theta }......\left(1\right) \end{gathered}$$

Here, $\theta$ is the angle between vector A and vector B. The magnitude of the resultant will be maximum if the value of angle $\theta$ is $0°$.

Substitute $\theta =0°$ values in the equation (1).

$$\begin{gathered} R_{\max }=\sqrt{A^2+B^2+2AB\left(\cos 0°\right)} \\ {R}_{\max }=\sqrt{A^2+B^2+2AB} \\ {R}_{\max }=\sqrt{{\left(A+B\right)}^2} \\ {R}_{\max }=A+B \end{gathered}$$

The relative direction of vectors A and B should be role="math" $0°$ for the maximum magnitude of the resultant.

Determination of resultant for minimum magnitude

Substitute $\theta =180°$ values in the equation (1) for the minimum magnitude of the resultant.

$$\begin{gathered} R_{\min }=\sqrt{A^2+B^2+2AB\left(\cos 180°\right)} \\ {R}_{\min }=\sqrt{A^2+B^2-2AB} \\ {R}_{\min }=\sqrt{{\left(A-B\right)}^2} \\ {R}_{\min }=A-B \end{gathered}$$

The relative direction of vectors A and B should be opposite to each other for the minimum magnitude of the resultant.

Therefore, the relative direction for maximum magnitude $\left(A+B\right)$should be in the same direction and the relative direction for minimum magnitude $\left(A-B\right)$ should be in the opposite direction.