If D, E, F are the midpoints of the sides BC, | vedclass

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(D) Let the position vectors of vertices $A, B, C$ be $\vec{a} = i + j$,$\vec{b} = j + k$,and $\vec{c} = k + i$.The position vectors of the midpoints

Vedclass · Accepted Answer

$\frac{2}{3}(i + j + k)$

Vedclass · Answer

(D) Let the position vectors of vertices $A, B, C$ be $\vec{a} = i + j$,$\vec{b} = j + k$,and $\vec{c} = k + i$.The position vectors of the midpoints $D, E, F$ are:$\vec{d} = \frac{\vec{b} + \vec{c}}{2} = \frac{(j + k) + (k + i)}{2} = \frac{i + j + 2k}{2}$$\vec{e} = \frac{\vec{a} + \vec{c}}{2} = \frac{(i + j) + (k + i)}{2} = \frac{2i + j + k}{2}$$\vec{f} = \frac{\vec{a} + \vec{b}}{2} = \frac{(i + j) + (j + k)}{2} = \frac{i + 2j + k}{2}$The centroid $G$ of $\Delta DEF$ is given by $\vec{g} = \frac{\vec{d} + \vec{e} + \vec{f}}{3}$.$\vec{g} = \frac{1}{3} \left( \frac{i + j + 2k + 2i + j + k + i + 2j + k}{2} \right)$$\vec{g} = \frac{1}{3} \left( \frac{4i + 4j + 4k}{2} \right) = \frac{1}{3} (2i + 2j + 2k) = \frac{2}{3}(i + j + k)$.

If $D, E, F$ are the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$ respectively,find the position vector of the centroid of $\Delta DEF$,given that the position vectors of $A, B, C$ are $i + j, j + k, k + i$ respectively.

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