Question

If \(\overset{̄}{a}\) and \(\overset{̄}{b}\) have the same magnitude and angle between them is \(60^{\circ}\) and their scalar product is \(\frac{1}{2}\), then \(|\overset{̄}{a}|\) is ____

• A. \(2\)
• B. \(3\)
• C. \(7\)
• D. \(1\)

Hint:

\(\vec a\cdot\vec b = |a||b|\cos60^{\circ}\) with \(|a|=|b|\).

Answer 1

The Correct Option is D

Step 1: Key Formula:
\(\vec a\cdot\vec b = |\vec a||\vec b|\cos\theta\).

Step 2: Calculate:
Since \(|\vec a| = |\vec b|\), we get \(\vec a\cdot\vec b = |\vec a|^2\cos60^{\circ} = \frac12|\vec a|^2\).
This equals \(\frac12\), so \(|\vec a|^2 = 1\) and \(|\vec a| = 1\).

Step 3: Other options:
Values \(2\), \(3\) and \(7\) would make the dot product \(2\), \(4.5\) and \(24.5\), not \(\frac12\).

Final Answer:
The magnitude is \(1\), option (D). \[ \boxed{1} \]