If a - and b - are unit vectors- then the greatest value of 3 a -b -a -b - isa 2b 22c 4d None of these

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Byju's Answer

Standard XII

Mathematics

Domain

If a → and b ...

Question

If $\vec{a} and \vec{b}$ are unit vectors, then the greatest value of $\sqrt{3} |\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|$ is
(a) 2
(b) $2 \sqrt{2}$
(c) 4
(d) None of these

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Solution

(c) 4

$We have \sqrt{3} |\vec{a} + \vec{b}| + |\vec{a} - \vec{b}| = \sqrt{3} \times \sqrt{{|\vec{a}|}^{2} + {|\vec{b}|}^{2} + 2 |\vec{a}| |\vec{b}| \cos θ} + \sqrt{{|\vec{a}|}^{2} + {|\vec{b}|}^{2} - 2 |\vec{a}| |\vec{b}| \cos θ} = \sqrt{3} \times \sqrt{1^{2} + 1^{2} + 2 \times 1 \times 1 \cos θ} + \sqrt{1^{2} + 1^{2} - 2 \times 1 \times 1 \cos θ} (As \vec{a} and \vec{b} unit vectors) = \sqrt{3} \times \sqrt{2 + 2 \cos θ} + \sqrt{2 - 2 \cos θ} = \sqrt{3} \times \sqrt{2 (1 + \cos θ)} + \sqrt{2 (1 - \cos θ)} = \sqrt{3} \times \sqrt{2 \times 2 \cos^{2} \frac{θ}{2}} + \sqrt{2 \times 2 \sin^{2} \frac{θ}{2}} = 2 \sqrt{3} \cos \frac{θ}{2} + 2 \sin \frac{θ}{2} = 2 (\sqrt{3} \cos \frac{θ}{2} + \sin \frac{θ}{2}) = 2 \times 2 (\frac{\sqrt{3}}{2} \cos \frac{θ}{2} + \frac{1}{2} \sin \frac{θ}{2}) = 2 \times 2 (\sin \frac{π}{3} \cos \frac{θ}{2} + \cos \frac{π}{3} \sin \frac{θ}{2}) = 4 \sin (\frac{π}{3} + \frac{θ}{2}) Now, m aximum value of \sin α = 1 \Rightarrow Maximum value of \sin (\frac{π}{3} + \frac{θ}{2}) = 1 \Rightarrow Maximum value of 4 \sin (\frac{π}{3} + \frac{θ}{2}) = 4 ∴ Maximum value of \sqrt{3} |\vec{a} + \vec{b}| + |\vec{a} - \vec{b}| = 4$

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If a →and b →are unit vectors, then the greatest value of 3 a →+b →+ a →-b → is (a) 2 (b) 22 (c) 4 (d) None of these

If $\vec{a} and \vec{b}$ are unit vectors, then the greatest value of $\sqrt{3} |\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|$ is
(a) 2
(b) $2 \sqrt{2}$
(c) 4
(d) None of these