# How to find a vector perpendicular to plane ABC given A,B and C

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:

Does it matter which vectors I use?

Because my attempt has got the exact opposite sign.

• Hint: cross product of two vectors – Matti P. Apr 15 at 12:46
• @MattiP. I have the solution and it shows that: they use AB x AC but how do you know to use these two vectors? – jdog Apr 15 at 12:50
• 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 – Tojrah Apr 15 at 12:59
• Again, sign does not matter, the two vectors opposite to each other, along same line are perpendicular to a plane. – Tojrah Apr 15 at 13:16
• 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. – Adam Latosiński Apr 15 at 13:33