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3-1.Vectors
hard
જો $ |\vec A \times \vec B| = \sqrt 3 \vec A.\vec B $ હોય, તો $ |\vec A + \vec B| $ નું મૂલ્ય શું થાય?
A
$ {\left( {{A^2} + {B^2} + \frac{{AB}}{{\sqrt 3 }}} \right)^{1/2}} $
B
$ A + B $
C
$ {({A^2} + {B^2} + \sqrt 3 AB)^{1/2}} $
D
$ {({A^2} + {B^2} + AB)^{1/2}} $
(AIPMT-2004)
Solution
(d)$|\,\overrightarrow A \times \overrightarrow B |\, = \sqrt 3 (\overrightarrow A .\overrightarrow B )$
$AB\sin \theta = \sqrt 3 AB\cos \theta \Rightarrow $$\tan \theta = \sqrt 3 $ $⇒$ $\theta = 60^\circ $
Now $|\overrightarrow R |\, = \,|\overrightarrow A + \overrightarrow B |\, = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } $
$ = \sqrt {{A^2} + {B^2} + 2AB\left( {\frac{1}{2}} \right)} $
$ = {({A^2} + {B^2} + AB)^{1/2}}$
Standard 11
Physics