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Find the Value of A, If Four Points with Position Vectors 3hati + 6hatj+9hatk, Hati + 2hatj + 3hatk,2hati + 3hatj + Hatk and 4hati + 6hatj + Lambdahatk Are Coplanar - Mathematics

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Question

Find the value of λ, if four points with position vectors `3hati + 6hatj+9hatk`, `hati + 2hatj + 3hatk`,`2hati + 3hatj + hatk` and `4hati + 6hatj + lambdahatk` are coplanar.

Solution

`"AB" = -2hat"i" - 4 hat "j" - 6 hat "k"`

`vec "AC" = - hat "i" - 3 hat "j" - 8 hat "k"`

`vec "AD" = hat "i" + 0 hat "j" +(lambda - 9) hat k`

`"AB" .(vec "AC" xx vec "AD") = 0 = [vec "AB"   vec "AC"   vec "AD"] = 0`

`=> |(2,-4,-6),(-1,-3,-8),(1,0,lambda - 9)| = 0`

`=> [6(lambda - 9) + 31] - [18 + 4 (lambda - 9)] = 0`

`=> 6lambda - 54 + 32 - 18 + 4lambda - 36`

`=> 2lambda - 54 + 32 - 18 + 36 = 0`

`=> 2lambda = 4`

`lambda = 2`

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Find the Value of A, If Four Points with Position Vectors 3hati + 6hatj+9hatk, Hati + 2hatj + 3hatk,2hati + 3hatj + Hatk and 4hati + 6hatj + Lambdahatk Are Coplanar Concept: Scalar Triple Product of Vectors.
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