Question

The position vectors of the vertices of an equilateral triangle, whose orthocentre is at the origin, then?

• A. $a+b+c=0$
• B. $a^2=b^2+c^2$
• C. $a+b=c$
• D. None of these

Answer 1

The correct option is A

$a+b+c=0$

Explanation for the correct option:

Find the required relation between the position vectors of an equilateral triangle

Assume that, the position vectors of the vertices of an equilateral triangle are $a,b$ and $c$.

Since the position vector of the centroid of the equivalent triangle is given by: $\frac{a+b+c}{3}$.

It is given that the centroid is at the origin.

Therefore,

$$\begin{gathered} \frac{a+b+c}{3}=0 \\ \Rightarrow a+b+c=0 \end{gathered}$$

Therefore, if the position vectors of the vertices of an equilateral triangle are $a,b$ and $c$, whose orthocentre is at the origin, then $a+b+c=0$.

Hence, option $A$, $a+b+c=0$ is the correct answer.