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  4. Let veca, vecb and vecc be three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D be veca- vecb+ vecc, λ veca-3 vecb+4 vecc,- veca+2 vecb-3 vecc and 2 veca-4 vecb+6 vecc respectively. If A B, A C and A D are coplanar, then λ is equal to

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

JEE MainJEE Main 2023Vector Algebra

Solution:

$ \overline{A B}=(\lambda-1) \bar{a}-2 \bar{b}+3 \bar{c}$
$\overline{A C}=2 \bar{a}+3 \bar{b}-4 \bar{c} $
$ \overline{A D}=\bar{a}-3 \bar{b}+5 \bar{c} $
$\begin{vmatrix}\lambda-1 & -2 & 3 \\ -2 & 3 & -4 \\ 1 & -3 & 5\end{vmatrix}=0 $
$ \Rightarrow(\lambda-1)(15-12)+2(-10+4)+3(6-3)=0$
$\Rightarrow(\lambda-1)=1 \Rightarrow \lambda=2$