Let D and E be the midpoints of the sides AC | vedclass

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

Question

(D) Let the position vectors of $A, B, C$ with respect to the origin $O$ be $\vec{a}, \vec{b}, \vec{c}$ respectively.Given that $\vec{a}+2\vec{b}+3\ve

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(D) Let the position vectors of $A, B, C$ with respect to the origin $O$ be $\vec{a}, \vec{b}, \vec{c}$ respectively.Given that $\vec{a}+2\vec{b}+3\vec{c}=\vec{0}$.Since $D$ is the midpoint of $AC$,the position vector of $D$ is $\vec{d} = \frac{\vec{a}+\vec{c}}{2}$.Since $E$ is the midpoint of $BC$,the position vector of $E$ is $\vec{e} = \frac{\vec{b}+\vec{c}}{2}$.The area of $	riangle ODE$ is given by $	ext{Area} = \frac{1}{2} |\vec{d} 	imes \vec{e}|$.Substituting the values of $\vec{d}$ and $\vec{e}$:$	ext{Area} = \frac{1}{2} |(\frac{\vec{a}+\vec{c}}{2}) 	imes (\frac{\vec{b}+\vec{c}}{2})| = \frac{1}{8} |\vec{a} 	imes \vec{b} + \vec{a} 	imes \vec{c} + \vec{c} 	imes \vec{b} + \vec{c} 	imes \vec{c}|$.Since $\vec{c} 	imes \vec{c} = \vec{0}$,we have $	ext{Area} = \frac{1}{8} |\vec{a} 	imes \vec{b} + \vec{a} 	imes \vec{c} - \vec{b} 	imes \vec{c}|$.From the given equation $\vec{a}+2\vec{b}+3\vec{c}=\vec{0}$,taking the cross product with $\vec{b}$ gives $\vec{a} 	imes \vec{b} + 3(\vec{c} 	imes \vec{b}) = \vec{0} \Rightarrow \vec{a} 	imes \vec{b} = 3(\vec{b} 	imes \vec{c})$.Taking the cross product with $\vec{c}$ gives $\vec{a} 	imes \vec{c} + 2(\vec{b} 	imes \vec{c}) = \vec{0} \Rightarrow \vec{a} 	imes \vec{c} = 2(\vec{c} 	imes \vec{b})$.Substituting these into the area expression:$	ext{Area} = \frac{1}{8} |3(\vec{b} 	imes \vec{c}) + 2(\vec{c} 	imes \vec{b}) - (\vec{b} 	imes \vec{c})| = \frac{1}{8} |3(\vec{b} 	imes \vec{c}) - 2(\vec{b} 	imes \vec{c}) - (\vec{b} 	imes \vec{c})| = \frac{1}{8} |0| = 0$.

Let $D$ and $E$ be the midpoints of the sides $AC$ and $BC$ of a triangle $ABC$ respectively. If $O$ is an interior point of the triangle $ABC$ such that $\overrightarrow{OA}+2\overrightarrow{OB}+3\overrightarrow{OC}=\overrightarrow{0}$,then the area (in sq. units) of the triangle $ODE$ is

Similar Questions

If $\bar{a}$ and $\bar{b}$ are two vectors such that $|\bar{a}|=5$,$|\bar{b}|=12$ and $|\bar{a}-\bar{b}|=13$,then $|2\bar{a}+\bar{b}|=$

The vector $a + b$ bisects the angle between the vectors $a$ and $b$,if

If $\frac{\pi}{2} < \theta \leq \pi$ and $|\overline{a}|=5, |\overline{b}|=13, |\overline{a} \times \overline{b}|=25$,then the value of $\overline{a} \cdot \overline{b}$ is

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$,$|\vec{a}| = 3$,$|\vec{b}| = 5$,and $|\vec{c}| = 7$,find the angle between $\vec{a}$ and $\vec{b}$.

If $|\vec{a}|=\sqrt{26}$,$|\vec{b}|=7$ and $|\vec{a} \times \vec{b}|=35$,then $\vec{a} \cdot \vec{b}$ is-

Vedclass Test Series

Exam Paper Generator

Online Exam Module