Basics of Vectors

$ABCD$ is a parallelogram. Show that:

$2\overrightarrow{DC}-\overrightarrow{DB}=\overrightarrow{AC}$

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

The position vectors of two points $A$ and $B$ are $\hat{i}-\hat{j}$ and $\hat{j}+\hat{k}$ respectively.

Consider the following points :

$1.$ $\left(-1,-3, 1\right)$

$2.$ $\left(-1, 3, 2\right)$

$3.$ $\left(-2, 5, 3\right)$

What is the magnitude of $\overrightarrow{AB}$?

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?

$\overrightarrow{A}=-\sqrt{3} \hat{i}+\hat{j}$ is rotated ${90}^{\circ }$ clockwise. Find the new vector.

For any vector $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, with $10\left|a_i\right|<1,i=1,2,3$, consider the following statements:
$\left(A\right)$ : $\max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}\le \left|\overrightarrow{a}\right|$
$\left(B\right)$ : $\left|\overrightarrow{a}\right|\le 3\max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}$

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 unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ make an angle $\frac{\mathrm{\pi }}{6}$ and $\frac{2\mathrm{\pi }}{3}$ with $x-$axis respectively, then $\left|\overrightarrow{a}+\overrightarrow{b}\right|=?$

If $\left|\overrightarrow{b}\right|=10$, then the vector $\overrightarrow{b}$ is collinear with vector $2\sqrt{2}\hat{i}-\hat{j}+4\hat{k}$ is

Let the vectors $\overrightarrow{a}=\left(1+t\right)\hat{i}+\left(1-t\right)\hat{j}+\hat{k}$, $\overrightarrow{b}=\left(1-t\right)\hat{i}+\left(1+t\right)\hat{j}+2\hat{k}$ and $\overrightarrow{c}=t\hat{i}-t\hat{j}+\hat{k}$, $t\in R$ $\alpha ,\beta ,\gamma \in R$, be such that for $\alpha \overrightarrow{a}+\beta \overrightarrow{b}+\gamma \overrightarrow{c}=\overrightarrow{0}\Rightarrow \alpha =\beta =\gamma =0$, then the set of all values of $t$ is

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

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

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

In parallelogram $ABCD$ in the figure given below, which of the following represent collinear both not equal vectors?

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