How do you know which way to do cross product? I\'m doing Calculus 2 right now a

Question

How do you know which way to do cross product? I'm doing Calculus 2 right now and we're doing planes and vectors. For example:

Find parametric equations for the line through point (0,1,2) that is parallel to the plane x+y+z=5 and perpendicular to the line x=1+t, y=1-t, z=2t.

I have the solution, but looking at the solution it says cross product of <1,1,1> and a direction vector <1,-1,2> which would yield <3,-1,-2>. according to the right hand rule, we could also do <1,-1,2> cross <1,1,1>. The normal vector will just be pointing the other way. How do we know which way to do cross product? Illustrations are welcome.

Explanation / Answer

You can do cross product any way you like. Both cases will give correct answer.

<1,1,1> X <1,-1,2> = <3,-1,-2>

3x-y-2z = c1

put (0,1,2)

c1= 0-1-4 =-5

<1,-1,2>X<1,1,1> = <-3,1,2>

-3x+y+2z = c2

put (0,1,2)

c2= 0+1+4 =5

The value of constant automatically comes with opposite sign: c1 = -c2

So the equation comes out to be same

The whole equation gets multiplied by -1...(3x-y-2z = 5 is same as -3x+y+2z = -5)

all points satisfied by 3x-y-2z = 5 will also be satisfied by -3x+y+2z = -5

for eg.(A line with equation y=x is same as a line with equation -y = -x)

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