If vec a, vec b, vecc are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then

Question

If vec a, vec b, vecc are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then (A)  veca + vecb + vecc = vec0 (B)  veca 2= vecb2 + vecc2  (C)  veca + vecb = vecc  (D) None of these

Tardigrade Admin · Accepted Answer

The position vector of the centroid of the triangle is

\frac{a + b + c}{3}

. Since, the triangle is an equilateral, therefore the orthocentre coincides with the centroid and hence

\frac{a + b + c}{3} = 0 \Rightarrow a + b + c = 0

Q. If $a, b, c$ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then

$a + b + c = 0$

$a^{2} = b^{2} + c^{2}$

$a + b = c$

None of these

The position vector of the centroid of the triangle is $\frac{a + b + c}{3}$ . Since, the triangle is an equilateral, therefore the orthocentre coincides with the centroid and hence
$\frac{a + b + c}{3} = 0 \Rightarrow a + b + c = 0$

Q. If a,b,c are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then

a+b+c=0

a2=b2+c2

a+b=c