Consider the following in respect of the vectors \vec{a}=(0,1,1) and \vec{b}=(1,0,1) :<br>1. The number of unit vectors perpendicular to <strong>BOTH</strong> \vec{a} and \vec{b} is <strong>ONLY</strong> one.<br>2. The angle between the vectors is \frac{\pi}{3}<br>Which of the statements given above is/are correct?

  1. A. 1 <strong>ONLY</strong>
  2. B. 2 <strong>ONLY</strong> ✓
  3. C. <strong>BOTH</strong> 1 and 2
  4. D. Neither 1 nor 2

Correct Answer: B. 2 <strong>ONLY</strong>

Explanation

Statement 1: There are always two oppositely directed unit vectors perpendicular to a plane containing two non-parallel vectors (\pm\frac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}). Thus, statement 1 is false. Statement 2: \cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|} = \frac{0(1)+1(0)+1(1)}{\sqrt{2}\sqrt{2}} = \frac{1}{2}, which implies \theta = \frac{\pi}{3}. Statement 2 is true.

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