Vector Algebra

Given,  $P\left(1,2,3\right)\&Q\left(4,5,6\right)$

So,

$\begin{array}{l}\overrightarrow{PQ}=(4-1)\hat{i}+(5-2)\hat{j}+(6-3)\hat{k}\\ =3\hat{i}+3\hat{j}+3\hat{k}\end{array}$

Kindly go through the solution

The given vectors are

$\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}$

$\overrightarrow{b}=\widehat{-2i}+4\hat{j}+5\hat{k}$

$\overrightarrow{c}=\hat{i}-6\hat{j}-7\hat{k}$

The sum of the vector is

$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\left(a_1+a_2+a_3\right)\hat{i}+\left(b_1+b_2+b_3\right)\widehat{\hat{j}+\left(c_1+c_2+c_3\right)\hat{k}}$

$\begin{array}{l}=(1-2+1)\hat{i}+(-2+4-6)\hat{j}+(1+5-7)\hat{k}\\ =0.\hat{i}+(-4)\hat{j}+(-1)\hat{k}\\ =-4\hat{j}-\stackrel{}{k}\end{array}$

Let the vector with initial point P (2,1) and terminal point Q. (-5,7) can be shown as,

$\begin{array}{l}\overrightarrow{PQ}=(-5,-2)\hat{i}+(7,-1)\hat{j}\\ \overrightarrow{PQ}=-7\hat{i}+6\hat{j}\end{array}$

The scalar components are -7 and 6.

The vector components are -7i and 6j.

Note that two vector are equal only if their corresponding components are equal.

Thus, the given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ will be equal if and only if $x=2\&y=3$

Two different vectors having same magnitude: -

(i) $2\hat{i}+\hat{j}+3\hat{k}$

(ii) $\hat{i}+3\hat{j}+2\hat{k}$

Kindly go through the solution

(i) True, as vector quantity $\overrightarrow{a}$ and - $\overrightarrow{a}$ are parallel to same line.

(ii) False, as collinear vector are those vectors that are parallel to same line, but it is not necessary that they are equal also.

(iii) False, as two vectors having same magnitude may have different directions, so they are not collinear.

(iv) False, as two collinear vectors having same magnitude are not equal whey they are opposite in direction.

(a) Vector $\overrightarrow{a}$ and $\overrightarrow{d}$ are co initial same initial point.

(b) $\overrightarrow{b}$ and $\stackrel{?}{d}$ same magnitude & direction.

(c) $\overrightarrow{a}$ and $\overrightarrow{c}$ are collinear but not equal they are parallels their direction are not same.