Question

Express $r^{\to }$ = $9i+2j-7k$ as a linear combination of $a^{\to }$= $i-2j+k$ and $b^{\to }$= $i+2j-3k$

Answer 1

$\overrightarrow{r}=9\hat{i}+2\hat{j}-7\hat{k}$

Let $\overrightarrow{r}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}=\lambda (\hat{i}-2\hat{j}+\hat{k})+\mu (\hat{i}+2\hat{j}-3\hat{k})=\hat{i}(\lambda +\mu )+\hat{j}(-2\lambda +2\mu )+\hat{k}(\lambda -3\mu )$

$\Rightarrow 9\hat{i}+2\hat{j}-7\hat{k}=\hat{i}(\lambda +\mu )+\hat{j}(-2\lambda +2\mu )+\hat{k}(\lambda -3\mu )$

Comparing L.H.S and R.H.S

$\Rightarrow \lambda +\mu =9\to \left(1\right)$

$\Rightarrow -2(\lambda -\mu )=2$

$\Rightarrow \lambda -\mu =-1\to \left(2\right)$

From $\left(1\right)$ and$\left(2\right)$, we get

$\lambda =4and\mu =4+1=5$

$-7=\lambda -3\mu$ must also be satisfied but $\lambda -3\mu =4-15=-9$

Hence, $\overrightarrow{r}$ can't be expressed as a linear combination of $\overrightarrow{a}$ and $\overrightarrow{b}$