Which one of the following relations is true for two unit vector A ^ a n d B ^ making an angle to each other?

| A ^ + B ^ | = A 2 + B 2 + 2 A B c o s θ = 1 + 1 + 2 c o s θ = 2 ( 1 + c o s θ ) = 2 * 2 c o s 2 θ 2 | A ^ + B ^ | = 2 c o s θ 2 -(1) | A ^ − B ^ | = A 2 + B 2 − 2 A B c o s θ = 1 + 1 − 2 c o s θ = 2 s i n θ 2 -(2) (2) ÷ (1) | A ^ − B ^ | | A ^ + B ^ | = 2 s i n θ 2 2 c o s θ 2 ⇒ | A ^ − B ^ | = | A ^ + B ^ | t a n θ 2

Which one of the following relations is true for two unit vector $\hat{A} a n d \hat{B}$ making an angle to each other?

Option 1 - <math> <mrow> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>A</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>B</mi> <mo>^</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>A</mi> <mo>^</mo> </mover> <mo>−</mo> <mover accent="true"> <mi>B</mi> <mo>^</mo> </mover> </mrow> <mo>|</mo> </mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> </mrow> </math> θ/2

Option 2 - <math> <mrow> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>A</mi> <mo>^</mo> </mover> <mo>−</mo> <mover accent="true"> <mi>B</mi> <mo>^</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>A</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>B</mi> <mo>^</mo> </mover> </mrow> <mo>|</mo> </mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> </mrow> </math> θ/2

Option 3 - <math> <mrow> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>A</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>B</mi> <mo>^</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>A</mi> <mo>^</mo> </mover> <mo>−</mo> <mover accent="true"> <mi>B</mi> <mo>^</mo> </mover> </mrow> <mo>|</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> </mrow> </math> θ/2

Option 4 - <math> <mrow> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>A</mi> <mo>^</mo> </mover> <mo>−</mo> <mover accent="true"> <mi>B</mi> <mo>^</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>A</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>B</mi> <mo>^</mo> </mover> </mrow> <mo>|</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> </mrow> </math> θ/2

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Raj Pandey | Contributor-Level 9

5 months ago

Correct Option - 2

Detailed Solution:

$| \hat{A} + \hat{B} | = \sqrt[]{A^{2} + B^{2} + 2 A B c o s θ}$

$= \sqrt[]{1 + 1 + 2 c o s θ}$

$= \sqrt[]{2 (1 + c o s θ)}$

= $\sqrt[]{2 * 2 c o s^{2} \frac{θ}{2}}$

$| \hat{A} + \hat{B} | = 2 c o s \frac{θ}{2}$ -(1)

$| \hat{A} - \hat{B} | = \sqrt[]{A^{2} + B^{2} - 2 A B c o s θ}$

$= \sqrt[]{1 + 1 - 2 c o s θ}$

$= 2 s i n \frac{θ}{2}$ -(2)

(2) $\div$ (1)

$\frac{| \hat{A} - \hat{B} |}{| \hat{A} + \hat{B} |} = \frac{2 s i n \frac{θ}{2}}{2 c o s \frac{θ}{2}}$

$\Rightarrow | \hat{A} - \hat{B} | = | \hat{A} + \hat{B} | t a n \frac{θ}{2}$

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