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5.1 · Q20

Q.Show that the following points are collinear: A(3,2,−4)A(3,2,-4), B(9,8,−10)B(9,8,-10), C(−2,−3,1)C(-2,-3,1).

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Let A(3,2,−4),B(9,8,−10),C(−2,−3,1)A(3,2,-4),B(9,8,-10),C(-2,-3,1). Compute

AB→=B−A=(6,6,−6),AC→=C−A=(−5,−5,5).\overrightarrow{AB}=B-A=(6,6,-6),\qquad \overrightarrow{AC}=C-A=(-5,-5,5).

Since AB→=(6,6,−6)=−65(−5,−5,5)\overrightarrow{AB}=(6,6,-6)=-\tfrac65(-5,-5,5)... more simply, note AC→=(−5,−5,5)=−56(6,6,−6)=−56AB→\overrightarrow{AC}=(-5,-5,5)= -\tfrac56(6,6,-6)=-\tfrac56\overrightarrow{AB}, so AC→\overrightarrow{AC} is a scalar multiple of

AB→\overrightarrow{AB}. Since both vectors start at the common point AA, this makes AB→\overrightarrow{AB} and …

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