LinAlg4.2.20

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Find the value of lambda for which the four points with position vectors 3{\bar  {i}}-2{\bar  {j}}-{\bar  {k}},2{\bar  {i}}+3{\bar  {j}}-4{\bar  {k}},-{\bar  {i}}+{\bar  {j}}+2{\bar  {k}},4{\bar  {i}}+5{\bar  {j}}+\lambda {\bar  {k}}\, are coplanar.

Let A,B,C,D be the given points. Then A=3{\bar  {i}}-2{\bar  {j}}-{\bar  {k}},B=2{\bar  {i}}+3{\bar  {j}}-4{\bar  {k}},C=-{\bar  {i}}+{\bar  {j}}+2{\bar  {k}},D=4{\bar  {i}}+5{\bar  {j}}+\lambda {\bar  {k}}\,

Now {\bar  {AB}}=-{\bar  {i}}+5{\bar  {j}}-3{\bar  {k}},{\bar  {AC}}=-4{\bar  {i}}+3{\bar  {j}}+3{\bar  {k}},{\bar  {AD}}={\bar  {i}}+7{\bar  {j}}+(\lambda +1){\bar  {k}}\,

SinceA,B,C,D are coplanar [{\bar  {AB}}{\bar  {AC}}{\bar  {AD}}]=0\, which implies

{\begin{vmatrix}-1&5&-3\\-4&3&3\\1&7&\lambda +1\end{vmatrix}}\, which is eual to 1(15+9)-7(-3-12)+(\lambda +1)(-3+20)=0\,

24+105+17\lambda +17=0,\lambda =-{\frac  {146}{17}}\,


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