Additional Further Pure Vectors

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A plane contains the points A (3, 0, 2), B (1, -1, 1) and C (2, 3, -1).
Find the equation of the plane in the form r.n=d .


Ive just completed this question using the vector product as im supposed to. However, what is important with vector product it is ant-commutative.

As I always do I found AB and then BC; and then found the vector product
AB x BC to give me n.

However, the solutions show AB and BC calculated but the negative of AB is used.It is notweorthy that AB had all negative components so is this why the negative was used.

But also couldnt I have labelled the points in any order so taking AB x BC would depend on which point had been called A,which B etc. etc.

So my question is how do I know which two vectors to take the vector product between and in what order.


Detailed explanations much appreciated; am selfteaching and my teachers havent been any help when its come to FP3 :/

When you're working out the product of 2 vectors that meet at a point, it is always a good idea to have them directed in the same sense - either both into the point (AB and CB), or both away from the point (BA and BC).

If you're working out the dot product, this ensures you get the right angle (correct sign for the cosine!).,

If you're working out the cross-product, you want your vectors to form a right-handed set like the basic i, j, k vectors - imagine turning a corkscrew from the i to the j axis, then your thumb points up the k-axis which is the direction a corkscrew would travel in (assuming you're right-handed )

It doesn't matter whether you give the answer exactly as in the solutions, or with the negative of n and the negative of d. They are just different ways of writing the same thing. In fact r.(kn) = kd is the same plane for any value of k. Sometimes you may be asked to write n as a unit vector, so that there is a unique solution.

When it comes to the cross product, it really doesn't matter which vectors you choose so long as they are joined head-to-head or tail-to-tail. This is because their cross product gives a normal vector perpendicular to both vectors, and therefore the plane (since a plane is defined by 3 non-collinear points). This normal vector can be any scalar multiple of your calculated vector and the answer will still be correct (so long as you have indeed calculated the cross product correctly).