Question:
Consider the vectors \(\vec{a} = \hat{i} – 2\hat{j} + \hat{k}\) and \(b = 4\hat{i} – 4\hat{j} + 7\hat{k}\). What is the vector perpendicular to both the vectors?
सदिशों \(\vec{a} = \hat{i} – 2\hat{j} + \hat{k}\) और \(\vec{b} = 4\hat{i} – 4\hat{j} + 7\hat{k}\) पर विचार कीजिए। दोनों सदिशों के लंबवत सदिश क्या है?
Options:
- \(10\hat{i} – 3\hat{j} – 4\hat{k}\)
- \(-10\hat{i} – 3\hat{j} + 4\hat{k}\)
- \(-10\hat{i} + 3\hat{j} + 4\hat{k}\)
- \(10\hat{i} – 3\hat{j} + 4\hat{k}\)
- \(10\hat{i} – 3\hat{j} – 4\hat{k}\)
- \(-10\hat{i} – 3\hat{j} + 4\hat{k}\)
- \(-10\hat{i} + 3\hat{j} + 4\hat{k}\)
- \(10\hat{i} – 3\hat{j} + 4\hat{k}\)
Correct Answer: c)
Explanation:
Vector perpendicular to both is \(\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 1 \\ 4 & -4 & 7 \end{vmatrix} = \hat{i}(-14+4) – \hat{j}(7-4) + \hat{k}(-4+8) = -10\hat{i} – 3\hat{j} + 4\hat{k}\) or \(-10\hat{i} + 3\hat{j} + 4\hat{k}\).
दोनों सदिशों के लंबवत सदिश \(\vec{a} \times \vec{b} = -10\hat{i} + 3\hat{j} + 4\hat{k}\) प्राप्त होता है।