LinAlg4.1.12

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Show that the points represented by {\bar  {i}}+2{\bar  {j}}+3{\bar  {k}},3{\bar  {i}}+4{\bar  {j}}+7{\bar  {k}},-3{\bar  {i}}-2{\bar  {j}}-5{\bar  {k}}\, are collinear.

Let A={\bar  {i}}+2{\bar  {j}}+3{\bar  {k}},B=3{\bar  {i}}+4{\bar  {j}}+7{\bar  {k}},C=-3{\bar  {i}}-2{\bar  {j}}-5{\bar  {k}}\, Let O be the origin,then

{\bar  {OA}}={\bar  {i}}+2{\bar  {j}}+3{\bar  {k}},{\bar  {OB}}=3{\bar  {i}}+4{\bar  {j}}+7{\bar  {k}},{\bar  {OC}}=-3{\bar  {i}}-2{\bar  {j}}-5{\bar  {k}}\,

Now {\bar  {AB}}={\bar  {OB}}-{\bar  {OA}}=(3{\bar  {i}}+4{\bar  {j}}+7{\bar  {k}})-({\bar  {i}}+2{\bar  {j}}+3{\bar  {k}})=2{\bar  {i}}+2{\bar  {j}}+4{\bar  {k}}\,

Similarly {\bar  {BC}}={\bar  {OC}}-{\bar  {OB}}=(-3{\bar  {i}}-2{\bar  {j}}-5{\bar  {k}})-(3{\bar  {i}}+4{\bar  {j}}+7{\bar  {k}})=-6{\bar  {i}}-6{\bar  {j}}-12{\bar  {k}}\,

Therefore {\bar  {BC}}=-3(2{\bar  {i}}+2{\bar  {j}}+4{\bar  {k}})=-3{\bar  {AB}}\,

|BC|=|-3|\cdot |AB|=3|AB|\,

AB and BC are two lines with common point B,hence A,B,C are collinear.


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