Let ABC be an acute scalene triangle,and O an | vedclass

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

Question

(C) Let the position vectors of the vertices of $	riangle ABC$ be $\vec{a}, \vec{b},$ and $\vec{c}$ respectively.Let the circumcentre $O$ of $	riang

Vedclass · Accepted Answer

$\frac{1}{2} \overrightarrow{HO}$

Vedclass · Answer

(C) Let the position vectors of the vertices of $	riangle ABC$ be $\vec{a}, \vec{b},$ and $\vec{c}$ respectively.Let the circumcentre $O$ of $	riangle ABC$ be the origin,so $\vec{O} = \vec{0}$.The position vector of the orthocentre $H$ is given by $\vec{H} = \vec{a} + \vec{b} + \vec{c}$.Since $N$ is the mid-point of $OH$,its position vector $\vec{n}$ is given by $\vec{n} = \frac{\vec{O} + \vec{H}}{2} = \frac{\vec{a} + \vec{b} + \vec{c}}{2}$.We need to find the sum $\overrightarrow{NA} + \overrightarrow{NB} + \overrightarrow{NC}$.This is equal to $(\vec{a} - \vec{n}) + (\vec{b} - \vec{n}) + (\vec{c} - \vec{n})$.$= (\vec{a} + \vec{b} + \vec{c}) - 3\vec{n}$.Substituting $\vec{n} = \frac{\vec{a} + \vec{b} + \vec{c}}{2}$,we get:$= (\vec{a} + \vec{b} + \vec{c}) - 3 \left( \frac{\vec{a} + \vec{b} + \vec{c}}{2} ight)$.$= (\vec{a} + \vec{b} + \vec{c}) \left( 1 - \frac{3}{2} ight) = -\frac{1}{2} (\vec{a} + \vec{b} + \vec{c})$.Since $\vec{H} = \vec{a} + \vec{b} + \vec{c}$,this is equal to $-\frac{1}{2} \vec{H} = -\frac{1}{2} \overrightarrow{OH} = \frac{1}{2} \overrightarrow{HO}$.

Let $ABC$ be an acute scalene triangle,and $O$ and $H$ be its circumcentre and orthocentre respectively. Further,let $N$ be the mid-point of $OH$. The value of the vector sum $\overrightarrow{NA}+\overrightarrow{NB}+\overrightarrow{NC}$ is

Similar Questions

Assertion $(A)$: The scalar product of force $\vec{F}$ and displacement $\vec{r}$ is equal to the work done.
Reason $(R)$: The work done is not a scalar quantity.

The value of $\lambda$ for which the vectors $2\lambda \hat{i} + \hat{j} - \hat{k}$ and $2\hat{j} + \hat{k}$ are perpendicular is:

If the position vectors of $A$ and $B$ are $6i + j - 3k$ and $4i - 3j - 2k$ respectively,then the work done by the force $\vec{F} = i - 3j + 5k$ in displacing a particle from $A$ to $B$ is ............ $units$.

If $a$,$b$,$c$ are the $p^{th}$,$q^{th}$,$r^{th}$ terms of an $A.P.$ and $\vec x = (q - r)\hat i + (r - p)\hat j + (p - q)\hat k$ and $\vec y = a\hat i + b\hat j + c\hat k$,then:

If $a, b$ and $c$ are three non-zero vectors,no two of which are collinear. If the vector $a + 2b$ is collinear with $c$ and $b + 3c$ is collinear with $a$,then ($\lambda$ being some non-zero scalar) $a + 2b + 6c$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module