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Question
Find the value of λ for which the four points with position vectors `6hat"i" - 7hat"j", 16hat"i" - 19hat"j" - 4hat"k" , lambdahat"j" - 6hat"k" "and" 2hat"i" - 5hat"j" + 10hat"k"` are coplanar.
Solution
Let the four points be A, B, C and D, whose position vectors are
P.V. of A = `6hat"i" - 7hat"j"` ,
P.V. of B = `16hat"i" - 19hat"j" - 4hat"k"`
P.V. of C = `lambdahat"j" - 6hat"k"` and
P.V. of D =`2hat"i" - 5hat"j" + 10hat"k"`
∴ `bar"AB" = 10hat"i" - 12hat"j" + 10hat"k"`
`bar"AC" = -6hat"i" + (lambda + 7)hat"j" - 6hat"k"`
`bar"AD" = -4hati + 2hat"j" + 10hat"k"`
Since A,B,C and D are coplanar
`=> bar"AB" , bar"AC" , bar"AD"` are coplanar
`=> bar"AB" bar "AC" bar"AD" = 0`
`=> |(10,-12,-4),(-6,lambda+7,-6),(-4,2,10)| = 0`
`=> 10(10lambda + 70 +12) + 12(-60-24) - 4(-12 + 4lambda + 28)= 0`
`=> 100lambda + 820 - 1008 - 64 - 16lambda = 0`
`=> 84lambda = 252`
`=> lambda = 3`
Henca the value of λ is 3.
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