Question

The points $\mathrm{A}(1,0,1),\mathrm{B}(–1,2,3)$ and $\mathrm{C}(0,1,2)$ are

• A. Non-collinear
• B. Collinear
• C. Non-coplanar
• D. None of the above

Answer 1

The correct option is B

Collinear

Identify the nature of the given points

Given : $\mathrm{A}(1,0,1),\mathrm{B}(–1,2,3)$ and $\mathrm{C}(0,1,2)$

Magnitude of $\overrightarrow{\mathrm{AB}}$

$$\begin{gathered} \overrightarrow{\left|AB\right|}=\sqrt{{(-1-1)}^2+{(2-0)}^2+{(3-1)}^2} \\ \Rightarrow \overrightarrow{\left|AB\right|}=\sqrt{4+4+4} \\ \Rightarrow \overrightarrow{\left|AB\right|}=2\sqrt{3} \end{gathered}$$

Magnitude of $\overrightarrow{\mathrm{BC}}$

$$\begin{gathered} \overrightarrow{\left|BC\right|}=\sqrt{{(0-1)}^2+{(1-2)}^2+{(2-3)}^2} \\ \Rightarrow \overrightarrow{\left|BC\right|}=\sqrt{1+1+1} \\ \Rightarrow \overrightarrow{\left|BC\right|}=\sqrt{3} \end{gathered}$$

Magnitude of $\overrightarrow{\mathrm{AC}}$

$$\begin{gathered} \overrightarrow{\left|AC\right|}=\sqrt{{(0-1)}^2+{(1-0)}^2+{(2-1)}^2} \\ \Rightarrow \overrightarrow{\left|AC\right|}=\sqrt{1+1+1} \\ \Rightarrow \overrightarrow{\left|AC\right|}=\sqrt{3} \end{gathered}$$

Since, $\left|\overrightarrow{AB}\right|=\left|\overrightarrow{BC}\right|+\left|\overrightarrow{AC}\right|$

$\Rightarrow$ $2\sqrt{3}=\sqrt{3}+\sqrt{3}$

Therefore, the point $C$ is the mid point of $AB$ then the points are collinear.

Hence, option $\left(B\right)$ is the correct answer.