For any two vectors $\vec{A}$ and $\vec{B}$ if $\vec{A} \cdot \vec{B}=|\vec{A} \times \vec{B}|$, the…
For any two vectors $\vec{A}$ and $\vec{B}$ if $\vec{A} \cdot \vec{B}=|\vec{A} \times \vec{B}|$, the magnitude of
$(\vec{A}+\vec{B})$ is $\left(\tan \frac{\pi}{4}=1, \cos \frac{\pi}{4}=\frac{1}{\sqrt{2}}\right)$
$\sqrt{A^{2}+B^{2}+\sqrt{2} A B}$
$\sqrt{A^{2}+B^{2}+\frac{A B}{\sqrt{2}}}$
$A+B$
$\sqrt{A^{2}+B^{2}}$
Solution
From the given condition,
$\mathrm{AB} \cos \theta=\mathrm{AB} \sin \theta$
So, $\theta=45^{\circ}$
Magnitude of $\mathrm{A}+\mathrm{B}$ will be,
$|\mathrm{A}+\mathrm{B}|=\sqrt{\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB} \cos 45^{\circ}}$
$=\sqrt{A^{2}+B^{2}+2 A B \times \frac{1}{\sqrt{2}}}$
$=\sqrt{A^{2}+B^{2}+\sqrt{2} A B}$
.