Question

Show that the points A, B, C with position vectors $\overrightarrow{a}-2\overrightarrow{b}+3\overrightarrow{c},2\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c}$ and $-7\overrightarrow{b}+10\overrightarrow{c}$ are collinear.

Answer 1

We have, A, B ,C with position vectors $\overrightarrow{a}-2\overrightarrow{b}+3\overrightarrow{c},2\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c},-7\overrightarrow{b}+10\overrightarrow{c}$ Then,
$\overrightarrow{AB}=$Position Vector of B $-$ Position Vector of A
$$\begin{gathered} =2\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c}-\overrightarrow{a}+2\overrightarrow{b}-3\overrightarrow{c} \\ =\overrightarrow{a}+5\overrightarrow{b}-7\overrightarrow{c} \end{gathered}$$

$\overrightarrow{BC}=$Position Vector of C $-$ Position Vector of B
$$\begin{gathered} =-7\overrightarrow{b}+10\overrightarrow{c}-2\overrightarrow{a}-3\overrightarrow{b}+4\overrightarrow{c} \\ =-2\overrightarrow{a}-10\overrightarrow{b}+14\overrightarrow{c} \\ =-2\left(\overrightarrow{a}+5\overrightarrow{b}-7\overrightarrow{c}\right) \end{gathered}$$
∴ $\overrightarrow{BC}=-2\overrightarrow{AB}$
Hence, $\overrightarrow{AB}\mathrm{and}\overrightarrow{BC}$ are parallel vectors.
But B is a point common to them.
So, $\overrightarrow{AB}\mathrm{and}\overrightarrow{BC}$ are collinear.
Hence, points A, B and C are collinear.