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Question
Show that the points A, B, C with position vectors \[\vec{a} - 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + 3 \vec{b} - 4 \vec{c}\] and \[- 7 \vec{b} + 10 \vec{c}\] are collinear.
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Solution
We have, A, B ,C with position vectors \[\vec{a} - 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + 3 \vec{b} - 4 \vec{c} , - 7 \vec{b} + 10 \vec{c}\]
Then,
\[\overrightarrow{AB} =\] Position Vector of B - Position Vector of A
\[= 2 \vec{a} + 3 \vec{b} - 4 \vec{c} - \vec{a} + 2 \vec{b} - 3 \vec{c} \]
\[ = \vec{a} + 5 \vec{b} - 7 \vec{c}\]
\[= - 7 \vec{b} + 10 \vec{c} - 2 \vec{a} - 3 \vec{b} + 4 \vec{c} \]
\[ = - 2 \vec{a} - 10 \vec{b} + 14 \vec{c} \]
\[ = - 2 \left( \vec{a} + 5 \vec{b} - 7 \vec{c} \right)\]
∴ \[\overrightarrow{BC} = - 2 \overrightarrow{AB}\]
Hence,
But B is a point common to them.
So,
Hence, points A, B and C are collinear.
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