Negative Volume using $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$

Question

So, my textbook explains how to find the volume of a paralelpiped using $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$. Makes sense, basically. But, when I go to do problems some combinations produce negative volumes. Example: $P(-2,1,0),\space Q(2,3,2),\space R(1,3,-1),\space S(3,6,1)$ and I compute the following vectors: $\mathbf{PQ}=<4,2,2>\space\mathbf{PR}=<3,3-1>\space\mathbf{PS}=<5,5,1>$.

However, this creates a little problem: $\mathbf{PR}\cdot(\mathbf{PQ}\times\mathbf{PS})=-16$ whereas $\mathbf{PQ}\cdot(\mathbf{PR}\times\mathbf{PS})=16$.

I know that the volume is 16, but where does the negative come from? What does it mean? How do I prevent, predict or use this behavoir?

Thanks!

Answer 1

The formula you are using gives in fact a signed volume, i.e. it can also be negative. The idea is the following: imagine for example a cube (with positive volume), now if you reduce the length of one of its edges, the volume will become smaller and smaller, until you get to $0$. Now, you can reduce the length of the edge further (making it become "negative", in some sense), and it makes sense that the volume would get smaller too, and thus negative. You can get the "correct", positive volume you're looking for simply by taking the absolute value of what you found, as SantiagoCanez remarked in the comments.

A side note: the most correct point of view for this is probably the one dealing with orientations, and obviously linear algebra but you'll learn about them only later (and probably only if you'll study maths or a related subject).

Answer 2

The cross product $B\times C$ doesn't quite the area, it gives the signed area of the parallelogram. What a signed area you ask? well, imaging you wanted to know the flow of water through a pipe - you don't only care about the cross sectional area, you also care about the direction of the flow. Then, if you don't care about the direction, you just take the absolute value to find the regular area.

Similarly, your formula gives the signed volume, since it's a product of the signed area with the height of the parallelepiped. To find the regular volume, just take the absolute value.

Answer 3

You can predict the behavior by the fact that if vectors $A, B, C$ in that order form a right-handed set of axes, then the formula will give a positive volume. If they form a left-handed set of axes (for example, $A = \hat{z}, B = \hat{y}, C = \hat{x} + \hat{y}$ is left-handed in that order) then the formula will give a negative volume.

It's hard to visualize the vectors; much easier ju cross and dot products and if it comes out negative, you know you specified a left-handed ordering.

Answer 4

Refer to the Right Hand Rule for triple products.

Basically, when you take the triple product of three vectors, you will get a signed measure of the volume of a parallelogram; it will be positive or negative depending on the order of multiplication. It's a matter of orientation.

Hold out your right hand so that the index finger (A) points forward, the ring finger (B) to your left, and the thumb points up (C). Anyway you rotate these vectors around the wrist (without changing the angles between them) will have the same triple product. Where as, the mirror symmetric representation on the left hand will have the negative of the right hand's triple product.

$$\begin{array}{cccc}\text{Right Hand} & \langle A, B, C\rangle = P & \langle C, A, B\rangle = P & \langle B, C, A\rangle = P \\ \text{Left Hand} & \langle A,C, B\rangle = - P & \langle C, B, A\rangle = -P & \langle B, A, C\rangle=-P \end{array}$$