91Ó°ÊÓ

$\stackrel{→}{a}-\stackrel{→}{b}=2\stackrel{→}{c}$…… (1)

$\stackrel{→}{a}+\stackrel{→}{b}=4\stackrel{→}{c}$…… (2)

$\stackrel{→}{c}=3\hat{i}+4\hat{j}$…… (3)

Using simple mathematical operations and laws of vector addition and subtraction, we can find the required answers.

Formula:

$\stackrel{→}{a}+\stackrel{→}{b}=\left(a_x+b_x\right)\stackrel{→}{i}+\left(a_y+b_y\right)\stackrel{→}{j}$

$\stackrel{→}{a}+\stackrel{→}{b}=\left(a_x-b_x\right)\stackrel{→}{i}+\left(a_y-b_y\right)\stackrel{→}{j}$

To determine the vector $\stackrel{→}{a}$, we need to add the equation (1) and (3) first.

We add the two equations in vectors as follows:

$\begin{array}{rcl}\left(\stackrel{→}{a}-\stackrel{→}{b}\right)+\left(\stackrel{→}{a}+\stackrel{→}{b}\right)& =& 2\stackrel{→}{c}+4\stackrel{→}{c}\\ 2\stackrel{→}{a}& =& 6\stackrel{→}{c}\\ \stackrel{→}{a}& =& 3\stackrel{→}{c}\\ & & \end{array}$

Substituting the value of $\stackrel{→}{c}$from equation (3),

$\begin{array}{rcl}\stackrel{→}{a}& =& 3\left(3\hat{i}+4\hat{j}\right)\\ & =& 9\hat{i}+12\hat{j}\\ & & \end{array}$

Therefore, the vector $\stackrel{→}{a}$is $9\hat{i}+12\hat{l}$.

To determine the magnitude of $\stackrel{\mathbf{→}}{\mathbf{b}}$, we need to subtract the equation (1) from (2).

$\begin{array}{rcl}\left(\stackrel{→}{a}+\stackrel{→}{b}\right)-\left(\stackrel{→}{a}-\stackrel{→}{b}\right)& =& 4\stackrel{→}{c}-2\stackrel{→}{c}\\ 2\stackrel{→}{b}& =& 2\stackrel{→}{c}\\ \stackrel{→}{b}& =& \stackrel{→}{c}\end{array}$

Substituting the value of$\stackrel{→}{c}$ from equation (3),

role="math" localid="1660890562947" $\stackrel{→}{b}=3\hat{i}+4\hat{j}$

Therefore, the vector $\stackrel{\mathbf{→}}{\mathbf{b}}$is $3\hat{i}+4\hat{j}$.