Scalars and Vectors

Explain dot product and cross product.

How do we find the angle between two vectors?

Define angle between two vectors?

Vector $A$has a magnitude of $10$ units and makes an angle of $30°$ with the positive x-axis. Vector $B$ has a magnitude of $20$ units and makes an angle of $30°$ with the negative x-axis. What is the magnitude of the resultant between these two vectors?

Define Localised Vector and give one example.

Define free vector and give one example.

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

Vector $A$has a magnitude of $10$ units and makes an angle of $30°$ with the positive x-axis. Vector $B$ has a magnitude of $20$ units and makes an angle of $30°$ with the negative x-axis. What is the magnitude of the resultant between these two vectors?

In a quadrilateral $PQRS$, $A$ divides $SR$ in the ratio $1: 3$ and $B$ is the mid point of $PR$. If $3\overrightarrow{SR}-\overrightarrow{QR}-3\overrightarrow{PS}-\overrightarrow{PQ}=k\overrightarrow{\mathrm{AB}},$ then $k=$

If a vector in the direction of $\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}$, which has a magnitude of $6 \text{units}$ is given by $x\hat{i}+y\hat{j}+z\hat{k}$, then find $x+y+z$.

Find the scalar and vector components of the vector with initial point $(2, 1)$ and terminal point $(–5, 7)$

Write two different vectors having same direction.

Compute the magnitude of the following vectors

$\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{b}=2\hat{i}-7\hat{j}-3\hat{k} , \overrightarrow{c}=\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}-\frac{1}{\sqrt{3}}\hat{k}$$$

Write two different vectors having same magnitude.

Prove that $2\hat{i}-\hat{j}+\hat{k}, \hat{i}-3\hat{j}-4\hat{k}$ and  $3\hat{i}-4\hat{j}-4\hat{k}$ are the vector sides of a right angle triangle.

Which of the following is not true if $\overrightarrow{A}=3\hat{i}+4\hat{j}$ and $\overrightarrow{B}=6\hat{i}+8\hat{j}$, where $A \& B$ are the magnitudes of $\overrightarrow{A}$ and $\overrightarrow{B}$? 

If $\left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=4$ and $\left|\overrightarrow{a}+\overrightarrow{b}\right|=5$, then $\left|\overrightarrow{a}-\overrightarrow{b}\right|=$

The vector $\hat{i}+x\hat{j}+3\hat{k}$ is rotated through an angle $\theta$ and doubled in magnitude, then it becomes $4\hat{i}+(4x-2)\hat{j}+2\hat{k}$. The values of $x$ are

A vector $\mathrm{A}$ makes an angle of ${20}^{\mathrm{o}}$ and $\mathrm{B}$ makes an angle of ${110}^{\mathrm{o}}$ with $x-$axis. The magnitudes of these vectors are $3 \mathrm{m}$ and $4 \mathrm{m}$ respectively. The resultant of the vectors is $k \mathrm{m}$. Find the value of $k$.