Let ABC be a triangle whose circumcentre is a | vedclass

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

Question

(C) Let $\vec{G}$ be the position vector of the centroid of $	riangle ABC$. We know that $\vec{G} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}$.Let $\vec{

Vedclass · Accepted Answer

$\frac{\vec{a} + \vec{b} + \vec{c}}{2}$

Vedclass · Answer

(C) Let $\vec{G}$ be the position vector of the centroid of $	riangle ABC$. We know that $\vec{G} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}$.Let $\vec{H}$ be the position vector of the orthocentre and $\vec{P}$ be the position vector of the circumcentre. We are given $\vec{P} = \frac{\vec{a} + \vec{b} + \vec{c}}{4}$.In any triangle,the centroid $\vec{G}$ divides the line segment joining the orthocentre $\vec{H}$ and the circumcentre $\vec{P}$ in the ratio $2:1$. That is,$\vec{G} = \frac{2\vec{P} + 1\vec{H}}{2+1} = \frac{2\vec{P} + \vec{H}}{3}$.Therefore,$3\vec{G} = 2\vec{P} + \vec{H}$,which implies $\vec{H} = 3\vec{G} - 2\vec{P}$.Substituting the values of $\vec{G}$ and $\vec{P}$:$\vec{H} = 3\left(\frac{\vec{a} + \vec{b} + \vec{c}}{3}ight) - 2\left(\frac{\vec{a} + \vec{b} + \vec{c}}{4}ight)$$\vec{H} = (\vec{a} + \vec{b} + \vec{c}) - \frac{\vec{a} + \vec{b} + \vec{c}}{2}$$\vec{H} = \frac{\vec{a} + \vec{b} + \vec{c}}{2}$.

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors of $A, B, C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac{\vec{a} + \vec{b} + \vec{c}}{4}$ respectively,then the position vector of the orthocentre of this triangle is:

Similar Questions

If $A(1, 2, 1), B(2, 3, 2), C(2, 1, 3), D(3, 2, 4)$,then the directions of $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are $.......$

Classify the following as scalar and vector quantities:
Force

If $\vec{a}$ is a nonzero vector of magnitude $a$ and $\lambda$ is a nonzero scalar,then $\lambda \vec{a}$ is a unit vector if

If $A \equiv (2i + 3j)$, $B \equiv (pi + 9j)$, and $C \equiv (i - j)$ are collinear, then the value of $p$ is: (in $/2$)

If the position vectors of points $A$ and $B$ are $2\hat{i} + 3\hat{j} + 4\hat{k}$ and $3\hat{i} - 4\hat{j} - 5\hat{k}$ respectively,find $\overline{AB}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module