Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 2: Level 2

If $P, Q, R$ are the mid-points of the sides $\overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CA}$, respectively of a $\Delta ABC$ and $\overrightarrow{a}, \overrightarrow{p}, \overrightarrow{q}$ are the position vector of $A, P, Q$, respectively, then what is the position vector of $R$?

In a $\Delta ABC, D, E, F$ are the mid-points of the sides $BC, CA$ and $AB$, respectively, the vector $\overrightarrow{AD}$ is equal to

If $D, E$ and $F$ are respectively mid-points of $AB, AC$ and $BC$ in $∆ABC$, then $\overrightarrow{BE}+\overrightarrow{AF}$ is equal to

Let $O$ be the origin. Let $\overrightarrow{OP}=x\hat{i}+y\hat{j}-\hat{k}$ and $\overrightarrow{OQ}=-\hat{i}+2\hat{j}+3x\hat{k}, x, y\in R, x>0,$ be such that $\left|\overrightarrow{PQ}\right|=\sqrt{20}$ and the vector $\overrightarrow{OP}$ is perpendicular to $\overrightarrow{OQ}.$ If $\overrightarrow{OR}=3\hat{i}+\mathrm{z}\hat{j}-7\hat{k}, z\in R,$ is coplanar with $\overrightarrow{OP}$ and $\overrightarrow{OQ},$ then the value of $x^2+y^2+z^2$ is equal to

Let $\overrightarrow{u}, \overrightarrow{v}$and $\overrightarrow{w}$ be such that $\left|\overrightarrow{u}\right|=1, \left|\overrightarrow{v}\right|=2$ and $\left|\overrightarrow{w}\right|=3$. If the projection of $\overrightarrow{v}$ along $\overrightarrow{u}$ is equal to that of $\overrightarrow{w}$ along $\overrightarrow{u}$ and vectors $\overrightarrow{v}$ and $\overrightarrow{w}$ are perpendicular to each other, then $\left|\overrightarrow{u}-\overrightarrow{v}+\overrightarrow{w}\right|$ equals:

If $\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}=\overrightarrow{0}$ and $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}=\lambda (\overrightarrow{b}\times \overrightarrow{c})$, then what is the value of $\lambda$?

If the magnitudes of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are equal, then which one of the following is correct?

Let three vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be such that $\overrightarrow{c}$ is coplanar with $\overrightarrow{a}$ and $\overrightarrow{b},\overrightarrow{a}·\overrightarrow{c}=7$ and $\overrightarrow{b}$ is perpendicular to $\overrightarrow{c},$ where $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}+\hat{k},$ then the value of $2{\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|}^2$ is