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

Question

1467 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-11T06:22:59

The unit vector is $=1/sqrt(2870)〈17,-30,-41〉$

First calculate the vector orthogonal to the other $2$ vectors. This is given by the cross product.

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

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

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

Therefore,

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

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

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

$=〈17,-30,-41〉=vecc$

Verification by doing 2 dot products

$〈17,-30,-41〉.〈-4,-5,2〉=(17)*(-4)+(-30)*(-5)+(-41)*(2)=0$

$〈17,-30,-41〉.〈-5,4,-5〉=(17)*(-5)+(-30)*(4)+(-41)*(-5)=0$

So,

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

The unit vector is

$hatc=vecc/(||vecc||)=1/sqrt(17^2+(-30)^2+(-41)^2)*〈17,-30,-41〉$

$=1/sqrt(2870)〈17,-30,-41〉$

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 (- 5 i + 4 j - 5 k) (- 5 i + 4 j - 5 k) ?