What is the unit vector that is orthogonal to the plane containing $(3i + 2j - 3k)$ and $( i - j + k)$ ?

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Answer 1 · 2016-02-29T07:52:28

$\hat{n}_{AB} = -1/\sqrt{62}(\hat{i}+6\hat{j}+5\hat{k})$

The unit vector perpendicular to the plane containing two vectors $\vec{A_{}}$ and $\vec{B_{}}$ is :

$\hat{n}_{AB} = \frac{\vec{A} \times \vec{B}} {|\vec{A} \times \vec{B}|}$

$\vec{A_{}} = 3\hat{i}+2\hat{j}-3\hat{k}; \qquad \vec{B_{}} = \hat{i}-\hat{j}+\hat{k};$

$\vec{A_{}}\times\vec{B_{}} = -(\hat{i}+6\hat{j}+5\hat{k});$
$|\vec{A_{}}\times\vec{B_{}}|=\sqrt{(-1)^2+(-6)^2+(-5)^2}=\sqrt{62}$

$\hat{n}_{AB} = -1/\sqrt{62}(\hat{i}+6\hat{j}+5\hat{k})$ .

What is the unit vector that is orthogonal to the plane containing (3i + 2j - 3k) (3i + 2j - 3k) and ( i - j + k) ( i - j + k) ?