If a - and b - are unit vectors- then write the value of a - b - 2- a - b - 2

It #160;is #160;given #160;that #160;a #8594; #160;and #160;b #8594; #160;are #160;unit #160;vectors. #8658;a #8594;=b #8594;=1 #1 ...

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Byju's Answer

Standard XII

Mathematics

Derivative of One Function w.r.t Another

If a → and b ...

Question

If $\vec{a} and \vec{b}$ are unit vectors, then write the value of ${|\vec{a} \times \vec{b}|}^{2} + {(\vec{a} . \vec{b})}^{2} .$

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Solution

$It is given that \vec{a} and \vec{b} are unit vectors. \Rightarrow |\vec{a}| = |\vec{b}| = 1 . . . (1) Now, {(\vec{a} . \vec{b})}^{2} + {|\vec{a} \times \vec{b}|}^{2} = {(|\vec{a}| |\vec{b}| \cos θ)}^{2} + {(|\vec{a}| |\vec{b}| \sin θ)}^{2} = {|\vec{a}|}^{2} {|\vec{b}|}^{2} (\cos^{2} θ + \sin^{2} θ) = {|\vec{a}|}^{2} {|\vec{b}|}^{2} (1) = {|\vec{a}|}^{2} {|\vec{b}|}^{2} = 1^{2} 1^{2} [From (1)] = 1$

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If a →and b →are unit vectors, then write the value of a →×b →2+ a →.b →2.

It is given that a→ and b→ are unit vectors.⇒a→=b→=1 ...(1)Now,a→. b→2+a→×b→2=a→ b→ cos θ2+a→ b→ sin θ2=a→2 b→2 cos2 θ+sin2 θ=a→2 b→2 1=a→2 b→2 =12 12 [From (1)]=1

If $\vec{a} and \vec{b}$ are unit vectors, then write the value of ${|\vec{a} \times \vec{b}|}^{2} + {(\vec{a} . \vec{b})}^{2} .$