The position vectors of the points A - B - C and D are 3 -2 -k- 2 -3 -2 k- 5 -2 k- and 4 -k- respectively- If the pointsA - B - C and D lie on a plane- the find the value of - -

Given, A=3 î 2 ĵ k̂ B=2 î 3 ĵ + 2 k̂ C=5 î ĵ + 2 k̂ D=4 î ĵ + λk̂ nbsp; Since, the points A, B, C and D lie on the plane, therefore vectors AB, AC, ...

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Byju's Answer

Standard XII

Mathematics

Perpendicular Distance of a Point from a Plane

The position ...

Question

The position vectors of the points $A, B, C$ and $D$ are $3^i - 2^j -^k, 2^i - 3^j + 2^k, 5^i -^j + 2^k$ and $4^i -^j + λ^k$ respectively. If the points $A, B, C$ and $D$ lie on a plane, the find the value of $λ$ ?

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Solution

Given, $\to A = 3^i - 2^j -^k$
$\to B = 2^i - 3^j + 2^k$
$\to C = 5^i -^j + 2^k$
$\to D = 4^i -^j + λ^k$

Since, the points $A, B, C$ and $D$ lie on the plane, therefore vectors $- - \to A B, - - \to A C, - - \to A D$ are coplanar.
$\Rightarrow - - \to A B \cdot (- - \to A C \times - - \to A D) = 0$
$- - \to A B = \to B - \to A = -^i -^j + 3^k - - \to A C = \to C - \to A = 2^i +^j + 3^k - - \to A D = \to D - \to A =^i +^j + (λ + 1)^k$

$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} - 1 & - 1 & 3 2 & 1 & 3 1 & 1 & 1 + λ \end{matrix} ∣ ∣ ∣ ∣ = 0$
$C_{1} \to C_{1} - C_{2}$
$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 0 & - 1 & 3 1 & 1 & 3 0 & 1 & 1 + λ \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow - λ - 4 = 0 \Rightarrow λ = - 4$

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The position vectors of the points A,B,C and D are 3^i−2^j−^k,2^i−3^j+2^k,5^i−^j+2^k and 4^i−^j+λ^k respectively. If the points A,B,C and D lie on a plane, the find the value of λ ?

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