What is the unit vector that is orthogonal to the plane containing $( - 4 i - 5 j + 2 k)$ and $(4 i + 4 j + 2 k)$ ?

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Answer 1 · 2018-05-13T05:35:00

The unit vector is $1/sqrt(596)*〈-18,16,4〉$

A vector that is orthogonal to $2$ other vectors is calculated with the cross product. The latter is calculate with the determinant.

$| (veci,vecj,veck), (d,e,f), (g,h,i) |$

where $veca=〈d,e,f〉$ and $vecb=〈g,h,i〉$ are the 2 vectors

Here, we have $veca=〈-4,-5,2〉$ and $vecb=〈4,4,2〉$

Therefore,

$| (veci,vecj,veck), (-4,-5,2), (4,4,2) |$

$=veci| (-5,2), (4,2) | -vecj| (-4,2), (4,2) | +veck| (-4,-5), (4,4) |$

$=veci((-5)*(2)-(4)*(2))-vecj((-4)*(2)-(4)*(2))+veck((-4)*(4)-(-5)*(4))$

$=〈-18,16,4〉=vecc$

Verification by doing 2 dot products

$〈-18,16,4〉.〈-4,-5,2〉=(-18)*(-4)+(16)*(-5)+(4)*(2)=0$

$〈-18,16,4〉.〈4,4,2〉=(-18)*(4)+(16)*(4)+(4)*(2)=0$

So,

$vecc$ is perpendicular to $veca$ and $vecb$

The unit vector is

$hatc=(vecc)/(||vecc||)$

The magnitude of $vecc$ is

$||vecc||=||〈-18,16,4〉||=sqrt((-18)^2+(16)^2+(4)^2)$

$=sqrt(596)$

The unit vector is $1/sqrt(596)*〈-18,16,4〉$

What is the unit vector that is orthogonal to the plane containing ( - 4 i - 5 j + 2 k) ( - 4 i - 5 j + 2 k) and (4 i + 4 j + 2 k) (4 i + 4 j + 2 k) ?