If a = \(\hat{i}+ \hat{j}+ \hat{k}\), b = \(\hat{j}- \hat{k}\), a x c = b, a.c = 3. Find [a, b, c]
(a) 2
(b) 6
(c) -2
(d) -6
Correct answer: (c) -2
Explanation:
\(\bar{a} = \hat{i}+ \hat{j}+ \hat{k}\), \(\bar{b} = \hat{j}- \hat{k}\)
Let \(\bar{c} = l\hat{i}+ m\hat{j} + n \bar{k}\)
\(\bar{b} \times \bar{c} = \bar{b}; \bar{a}.\bar{c}\) = 3
⟹ \((\bar{b} \times \bar{c})\times \bar{a}= \bar{b}\times\bar{a}\)
\((|\bar{a}|^2) \times \bar{c} -(\bar{a}. \bar{c})\bar{a}= \bar{b} \times \bar{a}\)
\(\bar{c} = \bar{a}+ \frac{1}{3} (\bar{b}\times \bar{a})\) ......(1)
∴ \(\bar{b} \times \bar{a}\) = \(\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\[0.3em]
0 & 1 & -1 \\[0.3em]
1 & 1 & 1
\end{vmatrix}\)
= \(2\hat{i}- \hat{j}- \hat{k}\)
∴ \(\bar{c} = (\hat{i}- \hat{j}- \hat{k})\) + \(\frac{1}{3}\) \((2\hat{i}- \hat{j}- \hat{k})\)
= \(\frac{5 \hat{i} + 2\hat{j} + 2\hat{k}}{3}\)
∴ \( \bar{a}.\bar{b}\;\bar{c}\) = \(\begin{vmatrix}
1 & 1 & 1 \\[0.3em]
0 & 1 & -1 \\[0.3em]
\frac{5}{3} & \frac{2}{3} & \frac{2}{3}
\end{vmatrix}\)
= \(\frac{-10}{3}\) + \(\frac{2}{3}\) + \(\frac{2}{3}\) = -2