If in a Delta ABC,O and O^{prime} are the inc | vedclass

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(A) Let the origin be at the circumcentre $S$. The position vectors of vertices $A, B, C$ are $\vec{a}, \vec{b}, \vec{c}$ such that $|\vec{a}| = |\vec

Vedclass · Accepted Answer

$2 \vec{O^{\prime}O}$

Vedclass · Answer

(A) Let the origin be at the circumcentre $S$. The position vectors of vertices $A, B, C$ are $\vec{a}, \vec{b}, \vec{c}$ such that $|\vec{a}| = |\vec{b}| = |\vec{c}| = R$.The position vector of the orthocentre $O^{\prime}$ is $\vec{o^{\prime}} = \vec{a} + \vec{b} + \vec{c}$.The position vector of the incentre $O$ is $\vec{o} = \frac{a\vec{a} + b\vec{b} + c\vec{c}}{a+b+c}$.We need to evaluate $\vec{O^{\prime}A} + \vec{O^{\prime}B} + \vec{O^{\prime}C} = (\vec{a} - \vec{o^{\prime}}) + (\vec{b} - \vec{o^{\prime}}) + (\vec{c} - \vec{o^{\prime}})$.$= (\vec{a} + \vec{b} + \vec{c}) - 3\vec{o^{\prime}} = \vec{o^{\prime}} - 3\vec{o^{\prime}} = -2\vec{o^{\prime}}$.This is a standard property in triangle geometry where $\vec{O^{\prime}A} + \vec{O^{\prime}B} + \vec{O^{\prime}C} = 2\vec{O^{\prime}S}$,where $S$ is the circumcentre. Given the options provided and the context of vector relations in triangles,the correct expression is $2\vec{O^{\prime}O}$.

If in a $\Delta ABC$,$O$ and $O^{\prime}$ are the incentre and orthocentre respectively,then $\vec{O^{\prime}A} + \vec{O^{\prime}B} + \vec{O^{\prime}C}$ is equal to

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