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