Question
Download Solution PDFIf \(\hat{a}\) and \(\hat{b}\) are unit vectors, and the angle between them is θ, then \(\sin \frac{\theta}{2}\) is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation:
\(\hat{a}\) and \(\hat{b}\) are unit vectors, and the angle between them is θ
So |\(\hat{a}\)| = 1, |\(\hat{b}\)| = 1 and \(\hat{a}\).\(\hat{b}\) = |\(\hat{a}\)||\(\hat{b}\)|cosθ = 1.1.cos θ = cos θ
Now, |\(\hat{a}\) - \(\hat{b}\)|2 = (\(\hat{a}\) - \(\hat{b}\))(\(\hat{a}\) - \(\hat{b}\)) = |\(\hat{a}\)|2 + |\(\hat{b}\)|2 - 2 \(\hat{a}\) . \(\hat{b}\)
⇒ |\(\hat{a}\) - \(\hat{b}\)|2 = 1 + 1 - 2cos θ
⇒ |\(\hat{a}\) - \(\hat{b}\)|2 = 2(1 - cos θ)
⇒ |\(\hat{a}\) - \(\hat{b}\)|2 = 2 × 2\(\sin^2\frac{\theta}{2}\)
⇒ |\(\hat{a}\) - \(\hat{b}\)|2 = 4\(\sin^2\frac{\theta}{2}\)
⇒ |\(\hat{a}\) - \(\hat{b}\)| = 2\(\sin \frac{\theta}{2}\) ⇒ \(\sin \frac{\theta}{2}\) = \(\frac{1}{2}|\hat{a}-\hat{b}|\)
(3) is correct
Last updated on May 25, 2025
-> BPSC Senior Secondary Teacher, BPSC TRE 4.0 is to be conducted in August, 2025.
-> CTET/STET-qualified candidates can appear for the BPSC TRE 4.0.
-> The Bihar Senior Secondary Teacher eligibility is PG + B.Ed./ B.El.ED + STET Paper-2 Pass.
-> The selection process includes a written exam. To boost your exam preparation, refer to the Bihar Senior Secondary Teacher Test Series.