The maximum and the minimum magnitude of the resultant two vectors are $17$ and $7$ units respectively. Then the magnitude of the resultant vector when they act perpendicular to each other is:

The correct option is D $13$

$\text{Step 1: Equation for resultant of two vectors}$

Let $\theta$ be the angle between the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$

The resultant of two vectors is given by

$R=\sqrt{A^2+B^2+2AB\cos \theta }$

For $R_{max},\theta =0^{\circ }$

$\Rightarrow |{\overrightarrow{R}}_{max}|=|\overrightarrow{A}|+|\overrightarrow{B}|$

$\Rightarrow 17=|\overrightarrow{A}|+|\overrightarrow{B}|$ $....\left(1\right)$

For $R_{min},\theta ={180}^{\circ }$

$\Rightarrow |{\overrightarrow{R}}_{min}|=|\overrightarrow{A}|-|\overrightarrow{B}|$

$\Rightarrow 7=|\overrightarrow{A}|-|\overrightarrow{B}|$ $....\left(2\right)$

$\text{Step 2: Calculation of Magnitude of}\overrightarrow{A}\text{and}\overrightarrow{B}$

Adding equation $\left(1\right)$ and $\left(2\right)$ $\Rightarrow 2|\overrightarrow{A}|=24$

$\Rightarrow \overrightarrow{A}=12$

From equation $\left(1\right)\Rightarrow$ $|\overrightarrow{B}|=5$

$\text{Step 3: Calculation of R when vectors are perpendicular}$

$\because \overrightarrow{A}\perp \overrightarrow{B}$

$|\overrightarrow{R}|=\sqrt{A^2+B^2+2AB\cos {90}^o}$

$\Rightarrow |\overrightarrow{R}|=\sqrt{{12}^2+5^2}=\sqrt{169}$ units

$\Rightarrow |\overrightarrow{R}|=13$ units

Hence the magnitude of resultant vector is $13$ units. Option $D$ is correct.