Question

Show that vectors $\hat{i}+\hat{j}+\hat{k},2\hat{i}+3\hat{j}-\hat{k},\widehat{-i}+2\hat{k}$ are linearly dependent.

Answer 1

If vector are linearly dependent then third vector can be written as any linear combination of two

$\therefore -\hat{i}+2\hat{k}=a(\hat{i}+\hat{j}+\hat{k})+b(2\hat{i}+3\hat{k}-\hat{k})$

$\therefore -\hat{i}=\hat{i}(a+2b)$

$\therefore o\hat{j}=\hat{j}(a+3b)$

$\therefore 2\hat{k}=\hat{i}(a-b)$

$\therefore -1=a+2b$

$-0=a+3b$

$-1=-b;b=1$

$a=-3b;a=-2$

Vectors are linear dependent.