Q. If $a,b,c$ are different real numbers and $a\widehat{\dot{\mathbf{i}}} + b\widehat{\dot{\mathbf{j}}} + c\widehat{\mathbf{k}},\ \ b\widehat{\dot{\mathbf{i}}} + c\widehat{\dot{\mathbf{j}}} + a\widehat{\mathbf{k}}$ and $c\widehat{\mathbf{i}} + a\widehat{j} + b\widehat{\mathbf{k}}\$are position vectors of three non-collinear points, then
A) $centroid\ of\ \mathrm{\Delta}ABC\ is\ \frac{a + b + c}{3}(\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}})$
B) $(\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}})$ is not really inclined to three vectors
C) Triangle $ABC$ is a scalene triangle
D) Perpendicular from the origin to the plane of the triangle does not meet it at the centroid
Explanation
(a)
The centroid of triangle
$= \frac{\left( a\widehat{\dot{\mathbf{i}}} + b\widehat{\dot{\mathbf{j}}} + c\widehat{\mathbf{k}} \right) + \left( b\widehat{\dot{\mathbf{i}}} + c\widehat{\dot{\mathbf{j}}} + a\widehat{\mathbf{k}} \right) + \left( c\widehat{\dot{\mathbf{i}}} + a\widehat{\dot{\mathbf{j}}} + b\widehat{\mathbf{k}} \right)}{3}$
$= \frac{a + b + c}{3}(\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}})$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.