Question

$ABCDEF$ is a regular hexagon where centre $O$ is the origin. If the position vectors of $A$ and $B$ are $\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k}$ and $2\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ respectively then $\overrightarrow{BC}$ is equal to

• A. $\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k}$
• B. $-\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k}$
• C. $3\overrightarrow{i}+3\overrightarrow{j}-4\overrightarrow{k}$
• D. None of these

Answer 1

Answer: (B)

$-\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k}$

Since its a regular hexagon, we need equal sides and so the modulus of $\overrightarrow{BC}$ has to be $\sqrt{6}$ and so option c gets eliminated.
Next, we can subtract the position vector of $B$ from the options to get the position vector of $C$.
Now checking that the angle between $\overrightarrow{B}$ and $\overrightarrow{C}$ to be ${60}^o$, we get option b to be correct.
$\overrightarrow{BC}=-\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k},\overrightarrow{C}=\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}$
Now $\overrightarrow{B}.\overrightarrow{C}=3$ and thus, the angle between $\overrightarrow{OB}$ and $\overrightarrow{OC}$ becomes ${60}^o$.