What is the unit vector that is orthogonal to the plane containing $( - 4 i - 5 j + 2 k)$ and $( i + 7 j + 4 k)$ ?

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Answer 1 · 2016-11-12T11:11:31

The unit vector is $=(1/sqrt2009)〈-34,18,-23〉$

We start by calculating the vector $vecn$ perpendicular to the plane.
We do a cross product
$=((veci,vecj,veck),(-4,-5,2),(1,7,4))$

$=veci(-20-14)-vecj(-16-2)+veck(-28+5)$

$vecn=〈-34,18,-23〉$

To calculate the unit vector $hatn$

$hatn=vecn/(∥vecn∥)$

$∥vecn∥=∥〈-34,18,-23〉∥=sqrt(34^2+18^2+23^2)=sqrt2009$

$hatn=(1/sqrt2009)〈-34,18,-23〉$

Let's do some checking by doing the dot product

$〈-4,-5,2〉.〈-34,18,-23〉=136-90-46=0$

$〈1,7,4〉.〈-34,18,-23〉=-34+126-92=0$

$:. vecn$ is perpendicular to the plane

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 ( i + 7 j + 4 k) ( i + 7 j + 4 k) ?