Question

Show that the points (3, 4), (−5, 16) and (5, 1) are collinear.

Answer 1

Let the given points be A(3, 4), B(−5, 16) and C(5, 1).
Now,
\[\overrightarrow{AB} = \left[ \left( - 5 \hat{i} + 16 \hat{j} \right) - \left( 3 \hat{i} + 4 \hat{j} \right) \right] = - 8 \hat{i} + 12 \hat{j}\]

\[\overrightarrow{AC} = \left[ \left( 5 \hat{i} + \hat{j} \right) - \left( 3 \hat{i} + 4 \hat{j} \right) \right] = 2 \hat{i} - 3 \hat{j}\]
Clearly,

\[\overrightarrow{AB} = - 4 \overrightarrow{AC}\]
Therefore,

\[\overrightarrow{AB}\] and \[\overrightarrow{AC}\] are parallel vectors. But, A is a common point of \[\overrightarrow{AB}\] and \[\overrightarrow{AC}\]
Hence, the given points (3, 4), (−5, 16) and (5, 1) are collinear.