What is the unit vector that is orthogonal to the plane containing $(29 i - 35 j - 17 k)$ $(29i-35j-17k)$ and $(41 j + 31 k)$ $(41j+31k)$ ?

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Answer 1 · 2017-04-02T10:18:52

The unit vector is $= \frac{1}{1540.3} ⟨ - 388, - 899, 1189 ⟩$ $=1/1540.3〈-388,-899,1189〉$

The vector perpendicular to 2 vectors is calculated with the determinant (cross product)

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

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

Here, we have $veca=〈29,-35,-17〉$ and $vecb=〈0,41,31〉$

Therefore,

$| (veci,vecj,veck), (29,-35,-17), (0,41,31) |$

$=veci| (-35,-17), (41,31) | -vecj| (29,-17), (0,31) | +veck| (29,-35), (0,41) |$

$=veci(-35*31+17*41)-vecj(29*31+17*0)+veck(29*41+35*0)$

$=〈-388,-899,1189〉=vecc$

Verification by doing 2 dot products

$〈-388,-899,1189〉.〈29,-35,-17〉=-388*29+899*35-17*1189=0$

$〈-388,-899,1189〉.〈0,41,31〉=-388*0-899*41+1189*31=0$

So,

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

The unit vector in the direction of $vecc$ is

$=vecc/||vecc||$

$||vecc||=sqrt(388^2+899^2+1189^2)=sqrt2372466$

The unit vector is $=1/1540.3〈-388,-899,1189〉$

What is the unit vector that is orthogonal to the plane containing (29i-35j-17k) (29i−35j−17k) (29i-35j-17k) and (41j+31k) (41j+31k) (41j+31k) ?