Which of the following relations is true for two unit vectors A and B making an angle θ to each other?

(A) |\hat{ A }+\hat{ B }|=|\hat{ A }-\hat{ B }| \tan \frac{\theta}{2}

(C) |\hat{ A }+\hat{ B }|=|\hat{ A }-\hat{ B }| \cos \frac{\theta}{2}

(D) |\overrightarrow{ A }-\hat{ B }|=|\overrightarrow{ A }+\hat{ B }| \cos \frac{\theta}{2}

JEE Mains · Physics · STD 11 - Vectors

Which of the following relations is true for two unit vectors \(\hat{ A }\) and \(\hat{ B }\) making an angle \(\theta\) to each other\(?\)

Pick an option to check your answer.

Verified Solution

Correct Answer

(B) \(|\hat{ A }-\hat{ B }|=|\hat{ A }+\hat{ B }| \tan \frac{\theta}{2}\)

Detailed explanation

\(|\hat{ A }+\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}+2|\hat{ A } \| \hat{ B }| \cos \theta}\)

\(=\sqrt{1+1+2 \cos \theta}\)

\(=\sqrt{2(1+\cos \theta)}\)

\(=\sqrt{2 \times 2 \cos ^{2} \frac{\theta}{2}}\)

\(=2 \cos \frac{\theta}{2}\)

\(|\hat{ A }-\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}-2|\hat{ A }||\hat{ B }| \cos \theta}\)

\(=\sqrt{2-2 \cos \theta}\)

\(=2 \sin \frac{\theta}{2}\)

\(\frac{|\hat{ A }+\hat{ B }|}{|\hat{ A }-\hat{ B }|}=\cot \frac{\theta}{2}\)

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