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

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