Question

Let $\overrightarrow{a}=1+4j+2k$ , $b=3i-2j+7k$ and $c=2i-j+4k$. Find a vector $\overline{d}\perp \overline{a}$ and $\overline{d}\perp \overline{b}$ and $\overrightarrow{c}\cdot \overrightarrow{d}=27$ .

Answer 1

Given $\overrightarrow{a}=\overrightarrow{i}+4\overrightarrow{j}+2\overrightarrow{k}$

$\overrightarrow{b}=3\overrightarrow{i}-2\overrightarrow{j}+7\overrightarrow{k}$

$\overrightarrow{c}=2\overrightarrow{i}-\overrightarrow{j}+4\overrightarrow{k}$

Let $\overrightarrow{d}=p\overrightarrow{i}+q\overrightarrow{j}+r\overrightarrow{k}$

Given $\overrightarrow{d}\perp \overrightarrow{a}$ and $\overrightarrow{d}\perp \overrightarrow{b}$

$\Rightarrow \overrightarrow{a}.\overrightarrow{d}=0\Rightarrow p+4q+2r=0$ ------- $\left(1\right)$

and $\overrightarrow{b}.\overrightarrow{d}=0\Rightarrow 3p-2q+7r=0$ ------ $\left(2\right)$

$\overrightarrow{c}.\overrightarrow{d}=27\Rightarrow 2p-q+4r=27$ ------ $\left(3\right)$

$\left(1\right)\Rightarrow p+4q+2r=0\Rightarrow 6p-4q+14r=0--------------7p+16r=0$ ---------- $\left(4\right)$

$\Rightarrow p+4q+2r=0x4\Rightarrow 8p-4q+16r=108-----------------9p+18r=108p+2r=18$ ------- $\left(5\right)$

$x8\Rightarrow 8p+16r=96x-1\Rightarrow -7p-16r=0-----------------p=96$

$2r=12-96$

$r=6-48=-42$

$p+4q+r=0$

$96-84+4q=0$

$4q=-12$

$q=-3$

Hence $\overrightarrow{d}=96\overrightarrow{i}-3\overrightarrow{j}-42\overrightarrow{k}$