Let and be three non-zero non-coplanar vectors. Let the position vector of four points A, B, C and D be and respectively. If and are coplanar, then is equal to ________ . [2023]

Question

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vector 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 $\vec{A B}, \vec{A C}$ and $\vec{A D}$ are coplanar, then $λ$ is equal to ________ . [2023]

Smart Achievers · Accepted Answer

(2)

$\vec{A B} = \vec{B} - \vec{A} = (λ - 1) \vec{a} - 2 \vec{b} + 3 \vec{c}$

$\vec{A C} = \vec{C} - \vec{A} = - 2 \vec{a} + 3 \vec{b} - 4 \vec{c}, \vec{A D} = \vec{a} - 3 \vec{b} + 5 \vec{c}$

$Since \vec{A B}, \vec{A C}, \vec{A D} are coplanar,$

So, $| \begin{matrix} λ - 1 & - 2 & 3 \\ - 2 & 3 & - 4 \\ 1 & - 3 & 5 \end{matrix} |$ $= 0$

$\Rightarrow (λ - 1) (15 - 12) + 2 (- 10 + 4) + 3 (6 - 3) = 0$

$\Rightarrow$ $λ = 2$

Solution

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Solution

Q. Let a→,b→ and c→ be three non-zero non-coplanar vectors. Let the position vector 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 ________ . [2023]

Ans. (2) AB→=B→-A→=(λ-1)a→-2b→+3c→ AC→=C→-A→=-2a→+3b→-4c→, AD→=a→-3b→+5c→ Since AB→, AC→, AD→ are coplanar, So, |λ-1-23-23-41-35|=0 ⇒(λ-1)(15-12)+2(-10+4)+3(6-3)=0 ⇒ λ=2

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