Question

Decompose the vector $\overrightarrow{b}=6\hat{i}-3\hat{j}-6\hat{k}$ into two vectors, $\overrightarrow{c}$ and $\overrightarrow{d}$, one of which is parallel and the other perpendicular to the vector $\overrightarrow{c}=\hat{i}+\hat{j}+\hat{k}$.

Answer 1

$\text{Given vector}6i-3j-6k=A$

$letA=c+d$

where c is paralle to $i+j+k$ and d is perpendicular to $i+j+k$ hence $c=m(i+j+k)$ 1 m-scalar

$\&d.(i+j+k)=0,d=a_1\hat{i}+a_2\hat{f}+a_3\hat{k}$

$\Rightarrow 6i-3j-3k=(m+a_1)\hat{i}+(m+a_2)\hat{j}+(m+a_3)\hat{k}$

$\text{equating},m+a_1=6$

$m+a_2=-3$

$m+a_3=-6$

$\text{But}{a}_1+a_2+a_3=0$

$\Rightarrow 3m=-3$

$\Rightarrow m=-1$

$\text{then}{a}_1=7,a_2=-2,a_3=-5$

$\Rightarrow 6i-3j-6k=-1(i+j+k)+(7i-2j-5k)$