Question:
Find the direction cosines of the vector joining the points A(1, 2, −3) and B(−1, −2, 1), directed from A to B.
बिंदुओं A(1, 2, −3) और B(−1, −2, 1) को मिलाने वाले, A से B की ओर निर्देशित सदिश की दिक्-कोज्याएँ (direction cosines) ज्ञात कीजिए।
Options:
- \(-\dfrac{1}{3}, \dfrac{2}{3}, \dfrac{2}{3}\)
- \(\dfrac{1}{3}, -\dfrac{2}{3}, \dfrac{2}{3}\)
- \(-\dfrac{1}{3}, -\dfrac{2}{3}, \dfrac{2}{3}\)
- \(-\dfrac{1}{3}, -\dfrac{2}{3}, -\dfrac{2}{3}\)
- \(-\dfrac{1}{3}, \dfrac{2}{3}, \dfrac{2}{3}\)
- \(\dfrac{1}{3}, -\dfrac{2}{3}, \dfrac{2}{3}\)
- \(-\dfrac{1}{3}, -\dfrac{2}{3}, \dfrac{2}{3}\)
- \(-\dfrac{1}{3}, -\dfrac{2}{3}, -\dfrac{2}{3}\)
Correct Answer: c)
Explanation:
\(\vec{AB} = (-1-1)\hat{i} + (-2-2)\hat{j} + (1-(-3))\hat{k} = -2\hat{i} – 4\hat{j} + 4\hat{k}\).
Magnitude \(= \sqrt{4+16+16} = 6\).
Direction cosines: \(\left(-\dfrac{2}{6}, -\dfrac{4}{6}, \dfrac{4}{6}\right) = \left(-\dfrac{1}{3}, -\dfrac{2}{3}, \dfrac{2}{3}\right)\).
\(\vec{AB} = -2\hat{i} – 4\hat{j} + 4\hat{k}\), परिमाण = 6।
दिक्-कोज्याएँ: \(\left(-\dfrac{1}{3}, -\dfrac{2}{3}, \dfrac{2}{3}\right)\)।