Discussion

The angle between two unit vectors \(\hat{A}\) and \(\hat{B}\) is 60° . The value of \(|\hat{A} - \hat{B}|\) is

A.\(\frac{1}{2}\)
B.\(\frac{3}{4}\)
C.\(\frac{1}{4}\)
D.1
E.\(\frac{1}{8}\)

Answer: D

\(|\hat{A} - \hat{B}|\) equals 1.

For unit vectors at angle \(\theta\),

\[ \begin{aligned} |\hat{A} - \hat{B}|^{2} &= 1 + 1 - 2\cos\theta \\ &= 2 - 2\cos 60^{\circ} = 1 \end{aligned} \]

so the magnitude is 1. Geometrically, the two unit vectors and their difference form an equilateral triangle, since the angle between them is 60°.

\(\frac{1}{2}\) is \(\cos 60^{\circ}\) itself, and the other fractions come from forgetting the square root or the factor 2.

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