Question

Find the magnitude of the vectors \(\vec a\) and \(\vec b\), if having equal magnitudes and angle \(60^\circ\) between them and their scalar product is \(\dfrac12\).

Hint:

Use \(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta\) with \(|\vec a|=|\vec b|\).

Answer 1

Step 1: Understanding the Concept:
The dot product formula \(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta\) connects the given scalar product, the (equal) magnitudes, and the angle.

Step 2: Setting up the equation:
Let \(|\vec a|=|\vec b|=k\). Then \(\vec a\cdot\vec b=k\cdot k\cdot\cos60^\circ=k^2\cdot\dfrac12\).

Step 3: Substituting the given value:
\(k^2\cdot\dfrac12=\dfrac12\ \Rightarrow\ k^2=1\ \Rightarrow\ k=1\) (taking the positive root, since magnitude can't be negative).

Final Answer:
\(|\vec a|=|\vec b|=\boxed{1}\).