Basics of Vectors

If $\lambda (2\hat{i}-4\hat{j}+4\hat{k})$ is a unit vector then $\lambda =?$

The position vectors of the vertices $A,B$ and $C$ of a triangle are $\hat{i}+\hat{j}, \hat{j}+\hat{k}$ and $\hat{i}+\hat{k}$ respectively. Find a unit vector lying in the plane of $ABC$ and perpendicular to $IA,$ where $I$ is the incentre of the triangle.

Given: $\overrightarrow{a}=\hat{i}+2\hat{j}$ and $\overrightarrow{b}=2\hat{i}+\hat{j}$, what are the magnitudes of the two vectors?Are these two vectors are equal?

If $\mathrm{a}=\overrightarrow{\mathrm{i}}+3\overrightarrow{\mathrm{j}}-\overrightarrow{\mathrm{k}}$ and $\mathrm{b}=2\overrightarrow{\mathrm{i}}+6\overrightarrow{\mathrm{j}}-\lambda \overrightarrow{\mathrm{k}}$. If $\mathrm{a} \mathrm{and} \mathrm{b}$ are parallel then find the value of $\lambda$.

In the following figure, which of the vectors are coinitial vectors?

In the following figure, which of the vectors are equal vectors.

Classify the following measure as scalars and vectors:  

$10 \mathrm{Newton}$

Classify the following measure as scalars and vectors:  

$20 \mathrm{m}/\mathrm{s}$ towards north

Classify the following measure as scalars and vectors:  

$10 \mathrm{gm}/{\mathrm{cm}}^3$

Classify the following measure as scalars and vectors:  

$1000 {\mathrm{cm}}^3$

Classify the following measure as scalars and vectors:  

$30 \mathrm{km}/\mathrm{hr}$

Classify the following measure as scalars and vectors:  

$5$ seconds

Shown below is a regular hexagon whose two vertices are joined by a vector.

Which of these statement(s) is/are true?

$\mathrm{i})$ $\overrightarrow{a} \mathrm{and} \overrightarrow{d}$ are equal vectors.

$\mathrm{ii}) \overrightarrow{b} \mathrm{and} \overrightarrow{e}$ are collinear vectors.

$\mathrm{iii}) \overrightarrow{c}, \overrightarrow{d} \mathrm{and} \overrightarrow{g}$ are coinitial vectors.

The position vectors of $X, Y \mathrm{and} \mathrm{Z}$ are $\hat{i}+4\hat{j}+3\hat{k},4\hat{j}+M\hat{j}+\frac{15}{2}\hat{k} \mathrm{and} 7\hat{\mathrm{i}}-4\hat{\mathrm{j}}+12\hat{\mathrm{k}}$. If these points are collinear, which could be the value of M.

 $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ are the position vectors of the points $A=(x,y,z) B=(y,-2z,3x) C=(2z,3x,-y)$ and $D=(1,-1,2)$ respectively. If $\left|\overrightarrow{a}\right|=2\sqrt{3},(\overrightarrow{a}^\overrightarrow{b})=(\overrightarrow{a}^\overrightarrow{c}), (\overrightarrow{a}^\overrightarrow{d})=\frac{\mathrm{\pi }}{2}$ and $(\overrightarrow{a}^\overrightarrow{j})$is obtuse, then find x, y, z

Find the magnitude of the vector $3\hat{i}-12\hat{j}+3\hat{k}$.

What are rectangular unit vectors$?$ Mention their usual notations.

Given $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(3\hat{i}-\hat{j}+\hat{k}\right)$ and $\overrightarrow{r}=\left(4\hat{i}-\hat{k}\right)+N\left(2\hat{i}+3\hat{k}\right)$. Find the point of intersection.

Find the angle between $\overrightarrow{A} \& \overrightarrow{B}, \overrightarrow{B} \& \overrightarrow{C}, \overrightarrow{C} \& \overrightarrow{A}.$

Angle made by a vector with $x,y \& z$ axes are in the ratio $1:2:3$. The angle made by the vector with $y-$axis is