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Scalars and Vectors

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Question

$If a = 2 \hat{i} - \hat{j} + \hat{k}, b = \hat{i} + 2 \hat{j} - 3 \hat{k} and c = 3 \hat{i} - y \hat{j} + 5 \hat{k} are coplanar. The value of y is _____$

Moderate

Solution

$If the vectors are coplanar. Then, a \cdot (b \times c) = 0 .$
$\begin{array}{l} \Rightarrow (2 \hat{i} - \hat{j} + \hat{k}) \cdot |\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & - 3 \\ 3 & - y & 5 \end{array}| = 0 \\ \Rightarrow (2 \tilde{i} - \tilde{j} + \tilde{k}) \cdot {\tilde{i} (10 - 3 y) - \tilde{j} (5 + 9) + \tilde{k} (- y - 6)} = 0 \\ \Rightarrow (2 \tilde{i} - \tilde{j} + \tilde{k}) \cdot {(10 - 3 y) \tilde{i} - 14 \tilde{j} - (6 + y) \tilde{k}} = 0 \\ \Rightarrow 2 (10 - 3 y) + 14 - (6 + y) = 0 \\ \Rightarrow 20 - 6 y + 14 - 6 - y = 0 \\ \Rightarrow 28 - 7 y = 0 ∴ y = 4 \end{array}$

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