Question

If$\overrightarrow{\mathbf{a}}\mathbf{-}\overrightarrow{\mathbf{b}}\mathbf{=}\mathbf{2}\overrightarrow{\mathbf{c}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{=}\mathbf{4}\overrightarrow{\mathbf{c}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{c}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{4}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$then what are (a)$\hat{\mathbf{a}}$and (b)$\hat{\mathbf{b}}$?

Short answer

1. The vector $\overrightarrow{a}$is role="math" localid="1660890164979" $9\hat{i}+12\hat{l}$.
2. The vector $\overrightarrow{b}$is $3\hat{i}+4\hat{j}$.

Step-by-step solution

Given data

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

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

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

Understanding the concept

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

Formula:

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

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

(a) Calculate the vector a⇀

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

We add the two equations in vectors as follows:

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

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

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

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

(b) Calculate the vector b→

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

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

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

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

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