If a vector of magnitude 2 units makes an angle \frac{\pi}{3} with 2\hat{i}, \frac{\pi}{4} with 3\hat{j} and an acute angle \theta with 4\hat{k}, then what are the components of the vector?

  1. A. (1,\sqrt{2},1) ✓
  2. B. (1,-\sqrt{2},1)
  3. C. (1,-\sqrt{2},-1)
  4. D. (1,\sqrt{2},-1)

Correct Answer: A. (1,\sqrt{2},1)

Explanation

The angles made with the coordinate axes are \alpha = \frac{\pi}{3}, \beta = \frac{\pi}{4}, and \gamma = \theta. The direction cosines are l = \cos\frac{\pi}{3} = \frac{1}{2} and m = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}. Using l^2+m^2+n^2=1, we get (\frac{1}{2})^2 + (\frac{1}{\sqrt{2}})^2 + n^2 = 1 \implies \frac{1}{4} + \frac{1}{2} + n^2 = 1 \implies n^2 = \frac{1}{4}. Since \theta is acute, n = \frac{1}{2}. The components of the vector with magnitude r=2 are (rl, rm, rn) = (2(\frac{1}{2}), 2(\frac{1}{\sqrt{2}}), 2(\frac{1}{2})) = (1, \sqrt{2}, 1).

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