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,

$\overrightarrow{\mathrm{AB}}=\left[\left(-5\hat{i}+16\hat{j}\right)-\left(3\hat{i}+4\hat{j}\right)\right]=-8\hat{i}+12\hat{j}$

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

Clearly, $\overrightarrow{\mathrm{AB}}=-4\overrightarrow{\mathrm{AC}}$

Therefore, $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$ are parallel vectors. But, A is a common point of $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$.

Hence, the given points (3, 4), (−5, 16) and (5, 1) are collinear.