Question

The position vectors of the vertices of an equilateral triangle, whose orthocenter is at the origin, then?
$A)a+b+c=0$
\[B){{a}^{2}}={{b}^{2}}+{{c}^{2}}\]
$C)a+b=c$
$D)$None of the above

Answer 1

Hint: To solve this question we need to have the knowledge of the property of equilateral triangles and also about the orthocenter. An equilateral triangle is a kind of triangle whose length of the three sides are equal. To solve this question the property used, will be that in an equilateral triangle the centroid and the orthocenter of the triangle coincides together. We will be applying the formula of centroid and will equate it with zero.

Complete step by step answer:
The question asks us to find the relation between the position vectors of the vertices of the equilateral triangle which are $a,b,c$ if the orthocenter coincides as the origin. For an equilateral triangle the most important theorem that we need to consider is that, in an equilateral triangle the centroid also coincides with the orthocenter so we will first find the centroid of the triangle and equate it to the orthocenter given which is the origin. so we know that the Centre centroid of the triangle is basically the ratio of the sum of the three vectors by $3$. On writing it mathematically we get:
$\Rightarrow \dfrac{a+b+c}{3}$
We will now equate the centroid to the orthocenter given in the question which is zero. On equating we get:
$\Rightarrow \dfrac{a+b+c}{3}=0$
$\Rightarrow a+b+c=0$
$\therefore $ The position vectors of the vertices of an equilateral triangle, whose orthocenter is at the origin, then $a+b+c=0$.

So, the correct answer is “Option A”.

Note: To solve this question we need to remember that in an equilateral triangle the orthocenter will always coincide with the centroid of the triangle which means both the points will have the same partition coordinate values.