Let $A=(1,0,1)$, $B=(2,1,-1)$, $C(0,1,2)$

Find a vector perpendicular to the plane $ABC$.

the solution I was given by my lecturer:

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Does it matter which vectors I use?

Because my attempt has got the exact opposite sign.

enter image description here

  • $\begingroup$ Hint: cross product of two vectors $\endgroup$ – Matti P. Apr 15 '19 at 12:46
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    $\begingroup$ You have to find a vector perpendicular to any two vectors in the plane. You can use AB and AC ,or AB and BC, or AC and BC $\endgroup$ – Tojrah Apr 15 '19 at 12:59
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    $\begingroup$ I want to emphasize that it could have been literally any two vectors so long as they were nonzero, linearly independent, and were both in the plane. There are infinitely many such choices. That they used $AB\times AC$ was merely personal preference on their part as they are some of the most apparent examples of vectors from the plane. $\endgroup$ – JMoravitz Apr 15 '19 at 13:05
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    $\begingroup$ Again, sign does not matter, the two vectors opposite to each other, along same line are perpendicular to a plane. $\endgroup$ – Tojrah Apr 15 '19 at 13:16
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    $\begingroup$ There's no single distinguished perpendicular vector, there's a whole 1-dimensional space of perpendicular vectors. If you find one such non-zero vector, you can obtain all the others by multiplying by various real numbers. Including -1. That's why the sign doesn't matter. Speaking of which, neither does the length, as long as it is non-zero. $\endgroup$ – Adam Latosiński Apr 15 '19 at 13:33

Your attempt is basically alright.

Take combination ABC and compute a unit normal vector P . Next take combination DBC and compute a unit normal vector Q.

In general, cross product M X N has sign opposite to that of N X M. The product vectors have opposite sense.

If P,Q are same then the sense is same, else the opposite sense has prevailed.

(If the vectors are altogether different then the four sides are not in the same plane.)

  • $\begingroup$ I'm not sure if I've fully understood your answer. Am I right in saying that the sign is basically given by the direction of the unit vector n in the equation: a×b = |a||b|sin(θ)n $\endgroup$ – jdog Apr 15 '19 at 13:44
  • $\begingroup$ The sign is given by the Right Hand Rule. The cross vector product of Thumb and Index fingers ..in that order..gives middle finger vector direction.of product .. When thumb/index fingers are swapped the sense changes to the opposite. $\endgroup$ – Narasimham Apr 15 '19 at 13:50

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