Addition of Vectors

Given that $\overrightarrow{P}+\overrightarrow{Q}+\overrightarrow{R}=\overrightarrow{0}.$ Two out of the three vectors are equal in magnitude. The magnitude of the third vector is $\sqrt{2}$ times that of the other two. Which of the following can be the angles between these vectors?

Three forces starts acting simultaneously on a particle moving with velocity $\overrightarrow{v}.$ These forces are represented in magnitude and direction by the three sides of a triangle $ABC$ (as shown). The particle will now move with velocity

If $\overrightarrow{P}+\overrightarrow{Q}=\overrightarrow{R}, \left|\overrightarrow{P}\right|=\left|\overrightarrow{Q}\right|=\left|\overrightarrow{R}\right|$, then angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ is

The position vectors of the vertices $\mathrm{P}, \mathrm{Q} \mathrm{and} \mathrm{R}$ of $∆\mathrm{PQR}$ are $-\hat{i}+2\hat{j}+4\hat{k}, 3\hat{i}+6\hat{j}+8\hat{k} \mathrm{and} 4\hat{i}+\hat{j}+\hat{k}$ respectively.

Which of the following is the vector that represents the median $\overrightarrow{\mathrm{PS}}$?

If $3\overrightarrow{A}=3\hat{i}+2\hat{j}$ and $\overrightarrow{B}=\hat{i}-2\hat{j}+3\hat{k}$. Then find the length of $\left(\overrightarrow{A}+\overrightarrow{B}\right)$ and $\left(\overrightarrow{A}-\overrightarrow{B}\right).$

If $A=3\hat{i}-4\hat{j}$ and $B=-\hat{i}-4\hat{j}$, then calculate $\overrightarrow{A}+\overrightarrow{B}$ and $\left|\overrightarrow{A}+\overrightarrow{B}\right|$.

Consider the two vector $\overrightarrow{A}$ and $\overrightarrow{B}$, the magnitude of their sum i.e $\left|\overrightarrow{A}+\overrightarrow{B}\right|$

When two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to each other, their magnitude of resultant is $R.$ When they are opposite to each other, their resultant is of magnitude $\frac{R}{\sqrt{2}}.$ Find the value of $\left(\frac{a}{b}+\frac{b}{a}\right).$

$ABCD$ is a tetrahedron and $O$ is any point. If the line joining $O$ to the vertices meet the opposite face at $P, Q, R$ and $S$, prove that $\frac{OP}{AP}+\frac{OQ}{BQ}+\frac{OR}{CR}+\frac{OS}{DS}=1$

In the triangle $PQR, \overline{PQ}=2\overline{a} \& \overline{QR}=2\overline{b}.$ The mid point of $PR$ is $M.$ FInd the vector $\overline{QM}$ in terms of $\overline{a}$ and $\overline{b}.$

If $\overrightarrow{A}=3\hat{i}-4\hat{j}-5\hat{k} \text{and} \overrightarrow{B}=2\hat{i}+6\hat{j}+8\hat{k}$ find $\overrightarrow{B}-\overrightarrow{A}$

If $\overrightarrow{A}=3\hat{i}-4\hat{j}-5\hat{k} \text{and} \overrightarrow{B}=2\hat{i}+6\hat{j}+8\hat{k}$ find $\overrightarrow{A}-\overrightarrow{B}$

If $\overrightarrow{A}=3\hat{i}-4\hat{j}-5\hat{k} \text{and} \overrightarrow{B}=2\hat{i}+6\hat{j}+8\hat{k}$ find $\overrightarrow{A}+\overrightarrow{B}$

In quadrilateral $ABCD, \overrightarrow{AB}=\overrightarrow{a}, \overrightarrow{BC}=\overrightarrow{b}, \overrightarrow{DA}=\overrightarrow{a}-\overrightarrow{b}, M$ is the midpoint of $BC$ and $X$ is a point on $DM$ such that $\overrightarrow{DX}=\frac{4}{5}\overrightarrow{DM}$. Then the points $A, X$ and $C$

If $\overrightarrow{A},\overrightarrow{B }\& \overrightarrow{C}$ represent unit vectors in each case, then which vector combination(s) results in a unit vector$?$

Find the magnitude of $2\overrightarrow{A}-3\overrightarrow{B}$ if $\overrightarrow{A}=\hat{i}+\hat{j}-2\hat{k} \text{and} \overrightarrow{B}=2\hat{i}-\hat{j}+\hat{k}$.

If $\overrightarrow{A},\overrightarrow{B }\& \overrightarrow{C}$ represent unit vectors in each case, then which vector combination(s) results in a unit vector$?$

Unit vectors along the diagonals of the parallelogram whose adjacent sides are $\hat{i}+\hat{j}+\hat{k} \& \hat{i}-\hat{j}+\hat{k}$, are

Let $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}, \overrightarrow{b}=5\hat{i}-3\hat{j}-3\hat{k}, \overrightarrow{c}=3\hat{i}-\hat{j}-2\hat{k}.$ If collinear with $\overrightarrow{c}$ and $\left|\overrightarrow{r}\right|=\left|\overrightarrow{a}+\overrightarrow{b}\right|.$ Then $\overrightarrow{r}$ equals