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.

Answer 1

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.