Question

If $\overline{a}=\hat{i}+\hat{j}-2\hat{k},\overline{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\overline{c}=3\hat{i}+\hat{k}$ and $\overline{c}=m\overline{a}+n\overline{b}$ then m + n =

• A. 0
• B. 1
• C. 2
• D. -1

Answer 1

Answer: (C)

2

We have,

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

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

$\overrightarrow{c}=3\hat{i}+\hat{k}$

And $\overrightarrow{c}=m\overrightarrow{a}+n\overrightarrow{b}$

Then,

From this equation,

$\overrightarrow{c}=m\overrightarrow{a}+n\overrightarrow{b}$

$3\hat{i}+\hat{k}=m(\hat{i}+\hat{j}-2\hat{k})+n(2\hat{i}-\hat{j}+\hat{k})$

$\Rightarrow 3\hat{i}+\hat{k}=m\hat{i}+m\hat{j}-2m\hat{k}+2n\hat{i}-n\hat{j}+n\hat{k}$

$\Rightarrow 3\hat{i}+\hat{k}=m\hat{i}+2n\hat{i}+m\hat{j}-n\hat{j}-2m\hat{k}+n\hat{k}$

$\Rightarrow 3\hat{i}+0\hat{j}+\hat{k}=(m+2n)\hat{i}+(m-n)\hat{j}-(2m+n)\hat{k}$

On comparing that,

$m+2n=3......\left(1\right)$

$m-n=0......\left(2\right)$

$2m+n=-1......\left(3\right)$

From equation (1) and (2) to, and we get,

$m=n=1$

So,

$m+n=1+1=2$

Hence, this is the answer.