Question

Check whether the vector $4\hat{i}-13\hat{j}+18\hat{k}$ is a linear combination of the vector $\hat{i}-2\hat{j}+3\hat{k}$ and $2\hat{i}+3\hat{j}-4\hat{k}$.

Answer 1

$4\hat{i}-13\hat{j}+18\hat{k}=a(\hat{i}-2\hat{j}+3\hat{k})+b(2\hat{i}+3\hat{j}-4\hat{k})$.

So, $a+2b=4-2a+3b=-13$

and $3a-4b=18$

On solving from any two, we at (basically first two here)

$a=\frac{38}{7},b=\frac{-5}{7}$

Since this does not satisfies $3a-4b=18$

We can say that linear combination is not possible.