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Find for which the points A(3, 2, 1), B(4, λ, 5), C(4, 2, – 2) and D(6, 5, – 1) are coplanar.

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Formula: -

(iii) Three vectors \(\vec a,\,\vec b\) and  \(\vec c\) are coplanar if and only if  \(\vec a.(\vec b\times\vec c)=0\)

let position vector of

\(OA=3\hat i+2\hat j+\hat k\)

position vector of

\(OB=4\hat i+\lambda\hat j+5\hat k\)

position vector of

\(OC=4\hat i+2\hat j-2\hat k\)

position vector of

\(OD=6\hat i+5\hat j-\hat k\)

The four points are coplanar if the vector \(\vec{AB},\vec {AC},\vec{AD}\) are coplanar

⇒ 1(9) – (λ – 2)( – 2 + 9) + 4(3 – 0) = 0

⇒ 7λ = 35

⇒ λ = 5

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