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Find λ, if ...

Question

Find $λ$ , if the vectors $\to a =^i + 3^j +^k, \to b = 2^i -^j -^k$ and $\to c = λ^j + 3^k$ are coplanar.

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Solution

For the three vectors to be coplanar the volume of the parallelopiped formed by them should be zero, so
$(\to a \times \to b) . \to c = 0 \Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 1 & 3 & 1 2 & - 1 & - 1 0 & λ & 3 \end{matrix} ∣ ∣ ∣ ∣ = - 21 + 3 λ = 0 \Rightarrow λ = 7.$

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