What is the value of \( r^2 \) ?

For the next two (02) items that follow : Let \( \vec{a}, \vec{b}, \vec{c} \) be unit vectors. Further, \( \vec{a} \) is perpendicular to \( \vec{b} \); \( \vec{c} \) makes an angle \( \frac{\pi}{3} \) with both \( \vec{a} \) and \( \vec{b} \); and \( \vec{c} = p\vec{a} + q\vec{b} + r(\vec{a} \times \vec{b}) \).

  1. A. 4
  2. B. 2
  3. C. 1
  4. D. \frac{1}{2} ✓

Correct Answer: D. \frac{1}{2}

Explanation

Since \vec{a}, \vec{b}, and \vec{a} \times \vec{b} are mutually orthogonal and have unit length, |\vec{c}|^2 = p^2 + q^2 + r^2. Given |\vec{c}| = 1, p = 1/2, q = 1/2, we get 1 = 1/4 + 1/4 + r^2, so r^2 = 1/2.

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