Question

If $\overline{a}=(2,1,-1),\overline{b}=(1,-1,0)$ , $\overline{c}=(5,-1,1)$, then the unit vector parallel to $\overline{a}+\overline{b}-\overline{c}$ but in the opposite direction is

• A. $-\frac{1}{3}(2\overline{i}-\overline{j}+2\overline{k})$
• B. $\frac{1}{3}(2\overline{i}-\overline{j}+2\overline{k})$
• C. $\frac{1}{3}(2\overline{i}+\overline{j}-2\overline{k})$
• D. $-\frac{1}{3}(2\overline{i}-\overline{j}-2\overline{k})$

Answer 1

The correct option is B $\frac{1}{3}(2\overline{i}-\overline{j}+2\overline{k})$

$\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}=i(2+1-5)+j(1-1+1)+k(-1+0-1)$
$=-2i+j-2k$
Unit vector along the above direction is
$=\frac{\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}}{|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}|}$
$=\frac{-2i+j-2k}{\sqrt{4+1+4}}$

$=\frac{-2i+j-2k}{3}$
Vector anti parallel to the above is
$=\frac{1}{3}(2i-j+2k)$