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

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

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

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

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

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Vector

Which of the following relations is true for two unit vectors

\hat{A}

and

\hat{B}

making an angle

θ

to each other?

Answer: (B)

For two unit vectors

\hat{A}

and

\hat{B}

at angle

θ

∣ \hat{A} + \hat{B} ∣ = 2 (1 + cos θ) = 2 cos \frac{θ}{2}

∣ \hat{A} - \hat{B} ∣ = 2 (1 - cos θ) = 2 sin \frac{θ}{2}

Taking the ratio:

\frac{∣ A ^ - B ^ ∣}{∣ A ^ + B ^ ∣} = \frac{2 sin \frac{θ}{2}}{2 cos \frac{θ}{2}} = tan \frac{θ}{2}

Therefore:

∣ \hat{A} - \hat{B} ∣ = ∣ \hat{A} + \hat{B} ∣ tan \frac{θ}{2}

Answer: (B)

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Single correct option