Let a = i +4 j +2 k , b =3 i -2 j +7 k and c =2 i - j +4 k . Find… - Vidyadip

Question

Let $\vec{\text{a }}=\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}},\vec{\text{b }}=3\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}\text{ and }\vec{\text{ c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$ Find a vector $\vec{\text{ p}}$ which is perpendicular to both $\vec{\text{a }}$ and $\vec{\text{ b}}$ and $\vec{\text{ p}}\cdot\vec{\text{ c}}$ = 18.

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Answer

$\vec{\text{p}}\text{ is }\bot\text{ }\text{ to both }\vec{\text{a}}\text{ and }\vec{\text{b}}\Rightarrow\vec{\text{p}}=\lambda\text{ }(\vec{\text{a}}\times\vec{\text{b}})$ Now $\vec{\text{a}}\times\vec{\text{b}}$ $\begin{vmatrix} \hat{\text{i}} & \hat{\text{j}} & \hat{\text{k}} \\ 1 & 4 & 2 \\ 3 & -2 & 7 \end{vmatrix}$ =$32\hat{\text{i}}-\hat{\text{j}}-14\hat{\text{k}}$ Given that $\vec{\text{ p}}\cdot\vec{\text{ c}}$ = 18 $\Rightarrow\lambda(32\hat{\text{i}}-\hat{\text{j}}-14\hat{\text{k}})\cdot(2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}})=18$ OR $\lambda(64+1-56)=18\Rightarrow\lambda=2$$\therefore\vec{\text{p}}=64\hat{\text{i}}-2\hat{\text{j}}-28\hat{\text{k}}$.

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