Q.20 Find the cartesian equation of the plane \(\bar{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})\).
Best answer
Answer: \(5x - 2y - 3z - 7 = 0\)
The plane passes through \(A(1, -1, 0)\) and is parallel to \(\bar{b} = \hat{i} + \hat{j} + \hat{k}\) and \(\bar{c} = \hat{i} - 2\hat{j} + 3\hat{k}\).
Normal: \(\bar{n} = \bar{b}\times\bar{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 1 & -2 & 3 \end{vmatrix} = 5\hat{i} - 2\hat{j} - 3\hat{k}\).
Equation: \(5(x - 1) - 2(y + 1) - 3(z - 0) = 0\), i.e. \(5x - 2y - 3z - 7 = 0\).
(Equivalently \(\begin{vmatrix} x - 1 & y + 1 & z \\ 1 & 1 & 1 \\ 1 & -2 & 3 \end{vmatrix} = 0\).)
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