What is ≤ft(θ/2) equal to?

Correct Answer is : ∣a^+b^∣2{|{a}+{b}|}{2}2∣a^+b^∣

Correct Answer is : ∣a^+b^∣2{|{a}+{b}|}{2}2∣a^+b^∣

Test Index

UPSC NDA Math Model Paper 9

Comprehension:(Que No. 60 - 61)

Direction: For the next two (2) items that follow:

Let

\hat{a},\hat{b}

be two unit vectors and θ be the angle between them

Question : 61 of 120

Marks: +1, -0

What is

\cos\left(\frac{\theta}{2}\right)

equal to?

Solution:

From previous question:

Given,

|\vec{a}|=1\,;\,|\vec{b}|=1

|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+|\vec{b}|^{2}+2(\vec{a}\cdot\vec{b})

=1+1+2(\vec{a}\cdot\vec{b})

=2+2|\vec{a}||\vec{b}|(\cos\theta)

=2(1+\cos\theta)

….(i)

|\vec{a}+\vec{b}|^{2}=2(1+\cos\theta)=4\cos^{2}\left(\frac{\theta}{2}\right)

\Rightarrow 4\cos^{2}\left(\frac{\theta}{2}\right)=|\vec{a}+\vec{b}|^{2}

\cos\left(\frac{\theta}{2}\right)=\sqrt{\frac{\left|\vec{a}+\vec{b}\right|^{2}}{4}}=\frac{\left|\vec{a}+\vec{b}\right|}{2}

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