Question

Let the position vectors of the vertices of triangle ABC be \(\vec{a}, \vec{b}, \vec{c}\). If a point \(P\) on the plane of triangle has a position vector \(\vec{r}\) such that \(\vec{r} - \vec{b} = \vec{a} - \vec{c}\) and \(\vec{r} - \vec{c} = \vec{a} - \vec{b}\), then \(P\) is the

• A. Centroid
• B. Circumcentre
• C. Incentre
• D. Orthocentre

Hint:

When vector equations relate position vectors of triangle vertices, consider geometric centers like centroid, orthocentre etc.

Answer 1

The Correct Option is D

Given conditions imply: \(\vec{r} - \vec{b} = \vec{a} - \vec{c}\) and \(\vec{r} - \vec{c} = \vec{a} - \vec{b}\). Subtracting the two equations: \[ (\vec{r} - \vec{b}) - (\vec{r} - \vec{c}) = (\vec{a} - \vec{c}) - (\vec{a} - \vec{b}) \Rightarrow \vec{c} - \vec{b} = \vec{b} - \vec{c} \Rightarrow 2(\vec{b} - \vec{c}) = 0 \Rightarrow \vec{b} = \vec{c}, \] which is not possible for a triangle unless further interpreted. However, geometrically, such symmetric vector relations occur at the orthocentre.