Let a¯,b¯ and c¯ be three non-zero coplanar vectors. Let the position vectors of four points A B C , , and D be a→−b→+c→,λa→−3b→+4c→, −a→+2b→−3c→ and 2a→−4b→+6c→ respectively. If AB→,AC→ and AD→ are coplanar, then λ is equal to ______.

AB¯AC¯AD¯=0⇒λ−1−23−23−41−35=0⇒(λ−1)3+2(−6)+3(3)=0⇒λ=2

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Let $\bar{a}, \bar{b} and \bar{c}$ be three non-zero coplanar vectors. Let the position vectors of four points A B C , , and D be $\vec{a} - \vec{b} + \vec{c}, λ \vec{a} - 3 \vec{b} + 4 \vec{c}, - \vec{a} + 2 \vec{b} - 3 \vec{c} and 2 \vec{a} - 4 \vec{b} + 6 \vec{c}$ respectively. If $\vec{AB}, \vec{AC} and \vec{AD}$ are coplanar, then $λ$ is equal to ______.

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answer is 2.

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Detailed Solution

$\begin{array}{l} [\begin{array}{lll} \bar{AB} & \bar{AC} & \bar{AD} \end{array}] = 0 \\ \Rightarrow |\begin{array}{ccc} λ - 1 & - 2 & 3 \\ - 2 & 3 & - 4 \\ 1 & - 3 & 5 \end{array}| = 0 \\ \Rightarrow (λ - 1) 3 + 2 (- 6) + 3 (3) = 0 \\ \Rightarrow λ = 2 \end{array}$