Question

if F1 vector and F2 vector are two vectors of equal magnitudes F vector such that |F1 vector •F2vector|=| F1 vector ×F2 vector| then |F1 vector+F2 vector| equal to​

Answer 1

$\vec{F_1}$ and $\vec{F_2}$ are vectors of equals magnitudes F such that $|\vec{F_1}.\vec{F_2}|=|\vec{F_2}\times\vec{F_2}|$

we know, magnitude of dot product is $|\vec{F_1}.\vec{F_2}|=|F_1||F_2|cos\theta$

and magnitude of cross product is $|\vec{F_1}\times\vec{F_2}|=|F_1||F_2|sin\theta$

so, now $|\vec{F_1}.\vec{F_2}|=|\vec{F_2}\times\vec{F_2}|$

or, $|F_1||F_2|cos\theta=|F_1||F_2|sin\theta$

given, $|F_1|=|F_2|=F$

or, $tan\theta=1$

then, $\theta=\frac{\pi}{4}$

now, $|F_1+F_2|=\sqrt{F_1^2+F_2^2+2F_1F_2cos\frac{\pi}{4}}$

given, $|F_1|=|F_2|=F$

so, $|F_1+F_2|=\sqrt{2F^2+\sqrt{2}F^2}$

=$\sqrt{2+\sqrt{2}}F$