Basics of Vectors

Three vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ represent the position vectors of the vertices of the triangle $ABC$ such that $\left|\overrightarrow{a}-\overrightarrow{d}\right|=\left|\overrightarrow{b}-\overrightarrow{d}\right|=\left|\overrightarrow{c}-\overrightarrow{d}\right|$, then the position vector of the orthocenter of the triangle is:

Position of a particle in a rectangular-coordinate system is $\left(3,2,5\right)$. Then its position vector will be

If direction cosines of a vector are $\frac{1}{\sqrt{5}}, \frac{-2}{\sqrt{5}}, \frac{1}{2\sqrt{5}}$ then it's vector equation is

In $∆OAC,$ if $B$ is the midpoint of side $AC$ and $\overrightarrow{OA}=\overrightarrow{a}, \overrightarrow{OB}=\overrightarrow{b}$ then $\overrightarrow{OC}=$

$M$ and $N$ are the mid points of the diagonals $AC$ and $BD$ respectively of quadrilateral $ABCD,$ then

$\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{CB}+\overrightarrow{CD}=$

A straight line which makes an angle of $60°$ with each of $y \mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }z$-axes, makes an angle with $x$-axis equal to

The direction cosines of the vector $3\hat{i}-4\hat{j}+5\hat{k}$ are

If $\overrightarrow{a}=\hat{i}+\hat{j}-2\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}-\hat{k}$. If $\overrightarrow{c}=m\overrightarrow{a}+n\overrightarrow{b}$ then $m+n=$

Calculate the magnitude of the given vector $\overrightarrow{a}=3\hat{i}+4\hat{j}+10\hat{k}$.

The angle between two adjacent sides $\overrightarrow{a}$ and $\overrightarrow{b}$ of parallelogram is $\frac{\pi }{6}$. If $\overrightarrow{a}=\left(2,-2,1\right)$ and $\left|\overrightarrow{b}\right|=2\left|\overrightarrow{a}\right|$, then area of this parallelogram is $\dots \dots$

If $\overrightarrow{x}$ is a vector in the direction of $\left(2,-2,1\right)$ of magnitude $6$ and $\overrightarrow{y}$ is a vector in the direction of $(1,1,-1)$ of magnitude $\sqrt{3}$, then $\left|\overrightarrow{x}+2\overrightarrow{y}\right|=\_\_\_\_\_$

Given three points $A, B\hspace{0.17em}\&\hspace{0.17em}C$ whose position vector are given as $2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c},\overrightarrow{a}-2\overrightarrow{b}+\lambda \overrightarrow{c}$and $\mu \overrightarrow{a}-5\overrightarrow{b}$. If the points $A, B \&\hspace{0.17em}C$ are collinear where  $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$are non-collinear, then 

Which triangle is formed by the points whose position vectors are $4\overrightarrow{i}-3\overrightarrow{j}+\overrightarrow{k}, 2\overrightarrow{i}-4\overrightarrow{j}+5\overrightarrow{k}, \overrightarrow{i}-\overrightarrow{j}$?

Which triangle is formed by the points whose position vectors are $4\overrightarrow{i}-3\overrightarrow{j}+\overrightarrow{k}, 2\overrightarrow{i}-4\overrightarrow{j}+5\overrightarrow{k}, \overrightarrow{i}-\overrightarrow{j}$?

A vector of magnitude $5$ units along the vector $\hat{i}-2\hat{j}+2\hat{k}$ is

If the vector $2\hat{i}-q\hat{j}+3\hat{k}$ and $4\hat{i}-5\hat{j}+6\hat{k}$ are colinear, find the value of $q$.

Let $\overrightarrow{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}},\overrightarrow{b}=2\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{c}=5\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ be three vectors. The area of the region formed by the set of points whose position vectors $\overrightarrow{r}$ satisfy the equations $\overrightarrow{r}·\overrightarrow{a}=5$  and $|\overrightarrow{r}-\overrightarrow{b}|+|\overrightarrow{r}-\overrightarrow{c}|=4$ is closest to the integer.

Find a value of $x$ for which $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector.