Answer to Question #324969 in Analytic Geometry for Agenda

Question

Find the unit vector that is perpendicular to vectors A=2i + j + k and B= -1 + 2j + k

Answer 1

$\hat{n} =\cfrac{\overrightarrow{A} \times\overrightarrow{B}} {|\overrightarrow{A} \times\overrightarrow{B}|}$

$\overrightarrow{A} \times\overrightarrow{B}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ 2 & 1 & 1\\ - 1 & 2 & 1 \end{vmatrix}=$

$=\hat{i}(1\cdot1-1\cdot2) -\hat{j}(2\cdot1-1\cdot(-1))+\\ +\hat{k}(2\cdot2-1\cdot(-1))=\\ =-\hat{i}-3\hat{j}+5\hat{k}$

$|\overrightarrow{A} \times\overrightarrow{B}|=\sqrt{(-1)^2+(-3)^2+5^2} =\sqrt{35}$

$\hat{n} =\cfrac{-\hat{i}} {\sqrt{35}} -\cfrac{3\hat{j}}{\sqrt{35}}+\cfrac{5\hat{k}}{\sqrt{35}}.$