If the position vectors of the vertices of a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the vertices of the triangle be $A, B, C$ with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively.The position vector of the centroid $

Vedclass · Accepted Answer

$\vec{0}$

Vedclass · Answer

(D) Let the vertices of the triangle be $A, B, C$ with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively.The position vector of the centroid $G$ is given by $\vec{g} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}$.The vectors from the vertices to the centroid are $\vec{GA} = \vec{a} - \vec{g}$,$\vec{GB} = \vec{b} - \vec{g}$,and $\vec{GC} = \vec{c} - \vec{g}$.The sum of these vectors is $\vec{GA} + \vec{GB} + \vec{GC} = (\vec{a} - \vec{g}) + (\vec{b} - \vec{g}) + (\vec{c} - \vec{g})$.$= (\vec{a} + \vec{b} + \vec{c}) - 3\vec{g}$.Substituting $\vec{g} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}$,we get:$= (\vec{a} + \vec{b} + \vec{c}) - 3 \left( \frac{\vec{a} + \vec{b} + \vec{c}}{3} \right) = (\vec{a} + \vec{b} + \vec{c}) - (\vec{a} + \vec{b} + \vec{c}) = \vec{0}$.

If the position vectors of the vertices of a triangle are $a, b, c$,then find the sum of the vectors from the vertices to the centroid.

Similar Questions

In a parallelogram $ABCD$,if the diagonals are $\overline{AC}$ and $\overline{BD}$,then $\overline{AC} - \overline{BD} = \dots$

In a regular hexagon $ABCDEF$,$\overrightarrow{AE} = $

The magnitude of the projection of the vector $\vec{a} = -\hat{i} + 2\hat{j} - \hat{k}$ on the unit vector $\hat{i}$ is . . . . . . .

If $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$,$\vec{b} = 2\hat{i} + 3\hat{j} + \hat{k}$,$\vec{c} = 3\hat{i} + \hat{j} + 2\hat{k}$ and $\alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} = -3(\hat{i} - \hat{k})$. Then the triplet $(\alpha, \beta, \gamma)$ is

In $\triangle OAC$,if $B$ is the mid-point of side $AC$ and $\vec{OA}=\vec{a}, \vec{OB}=\vec{b}$,then $\vec{OC}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module