Basics of Vectors

The magnitudes of vectors  $\overrightarrow{A},\overrightarrow{B}\mathrm{and}\overrightarrow{C}$  are $3,$ $4$ and $5$ units respectively. If  $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{C}$ , then the angle between  $\overrightarrow{A}\mathrm{and}\overrightarrow{B}$  is :

State the definition and value of zero vector in three-dimensional geometry. Find sum of zero vector and $2\hat{i}+4\hat{j}+6\hat{k}$.

State the definition and value of zero vector in three-dimensional geometry. Find sum of zero vector and $2\hat{i}+2\hat{j}+5\hat{k}$.

Define vector and find the sum of the following vectors.

$2\hat{i}+4\hat{j}+3\hat{k}$, $3\hat{i}+2\hat{j}+3\hat{k}$

Define vector and find the sum of the following vectors.

$2\hat{i}+3\hat{j}+5\hat{k}$, $3\hat{i}+4\hat{j}+6\hat{k}$

State the definition and value of zero vector in three-dimensional geometry. Find sum of zero vector and $2\hat{i}+3\hat{j}+4\hat{k}$.

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

Find the square of magnitude of the vector joining two points $\left(\sin x,\cos x,1\right)$ and $\left(-\cos x,\sin x,4\right)$ in $3D$.

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 the position vectors of the points which divides internally and externally in the ratio $2:3$ the join of points $2\overrightarrow{a}-3\overrightarrow{b}$ and $3\overrightarrow{a}-2\overrightarrow{b}$.

Find the value of $k$ so that the lines are at right angles.

$\frac{x-2}{2k}=\frac{y}{3}=\frac{z}{4}$ and $\frac{x}{3}=\frac{y+2}{k}=\frac{z-1}{-9}$

The angle between the two vector quantities $\overrightarrow{A}=4\widehat{i}-2\widehat{j}-\widehat{k}$ and $\overrightarrow{B}=\widehat{i}+4\widehat{j}-4\widehat{k}$ is equal to

Two equal forces act at a point. The square of their resultant is three times their product. What is the angle between them?

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be three unit vectors,$\overrightarrow{b}$ and $\overrightarrow{c}$ are non-parallel, such that $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)=\frac{\overrightarrow{b}+\overrightarrow{c}}{2},$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is:

If $\left|\overrightarrow{A}+\overrightarrow{B}\right|=n\left|\overrightarrow{A}-\overrightarrow{B}\right|$ and $\left|A\right|=\left|B\right|$ then what is angle between $\overrightarrow{A}$ and $\overrightarrow{B}$?

Cube of side $a$ is given in figure.


Equation of vector joining mid-point of surface in $x-z$ plane to mid-point of surface in $y-z$ plane.

The set of values of c for which the angle between the vectors $cx\widehat{i}-6\widehat{j}+3\widehat{k} and x\widehat{i}-2\widehat{j}+2cx\widehat{k}$ is acute for every $x\in R$ is

Suppose $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0, \left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=5, \left|\overrightarrow{c}\right|=7,$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is