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.

Answer 1

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 
\[= 2 \vec{a} + 3 \vec{b} - 4 \vec{c} - \vec{a} + 2 \vec{b} - 3 \vec{c} \]
\[ = \vec{a} + 5 \vec{b} - 7 \vec{c}\]

\[\overrightarrow{BC} =\] Position Vector of C - Position Vector of B
\[= - 7 \vec{b} + 10 \vec{c} - 2 \vec{a} - 3 \vec{b} + 4 \vec{c} \]
\[ = - 2 \vec{a} - 10 \vec{b} + 14 \vec{c} \]
\[ = - 2 \left( \vec{a} + 5 \vec{b} - 7 \vec{c} \right)\]
∴ \[\overrightarrow{BC} = - 2 \overrightarrow{AB}\]
Hence, 

\[\overrightarrow{AB} \text{ and }\overrightarrow{BC}\] are parallel vectors.
But B is a point common to them.
So,

\[\overrightarrow{AB}\text{ and }\overrightarrow{BC}\] are collinear.
Hence, points A, B and C  are collinear.