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

Answer 1

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 perform the dot product to check a few things.

#〈-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

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Answer 2

Xavier Adame

The unit vector orthogonal to the given plane is $\frac{1}{\sqrt{66}} (-2i - 14j - 7k)$ .

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.