How many of the following can be a vector perpendicular to <strong>BOTH</strong> the vectors 2\hat{i}-\hat{j}+\hat{k} and \hat{i}+\hat{j}+3\hat{k}?<br>I. 4\hat{i}+5\hat{j}-3\hat{k}<br>II. -8\hat{i}-10\hat{j}+6\hat{k}<br>III. \frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})<br>Select the correct answer.

  1. A. None
  2. B. One
  3. C. Two
  4. D. All three ✓

Correct Answer: D. All three

Explanation

A vector perpendicular to both \vec{u} and \vec{v} must be parallel to their cross product \vec{u} \times \vec{v}. Let \vec{u} = 2\hat{i}-\hat{j}+\hat{k} and \vec{v} = \hat{i}+\hat{j}+3\hat{k}. Then \vec{u} \times \vec{v} = \hat{i}(-3 - 1) - \hat{j}(6 - 1) + \hat{k}(2 - (-1)) = -4\hat{i} - 5\hat{j} + 3\hat{k}. Vectors I, II, and III are all scalar multiples of this vector (\times -1, \times 2, and \times \frac{1}{50} respectively). Hence, all three are perpendicular to the original vectors.

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