Order of operation for the Cross Product in linear algebra

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I realize that the order of operation for the Cross Product actually matters. I understand that AxB = -BxA. (when A and B are vectors)

But I'm confused in finding the correct order. If you take a look at the picture I linked below, the exercise decided to label PQ vector U, and PR vector V. After that, they used the Cross Product with the following order: UxV.

So, is there a reason for assigning PQ vector U, and PR vector V? Why not the other way? Why can't PQ be V, and PR be U?

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2 Answers

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In this case, a choice of sign does not matter, since the area of the triangle is given in terms of the norm of the cross product of two vectors. Remember that the norm of a vector is equal to the norm of the negative of that vector.

Note the solution in the image is missing a bit: It should be

$$\frac 1 2 \|u \times v \|$$

at the first step, not just $\frac 1 2 u \times v$. A similar caveat applies to the right hand side of the first equality. (Notice the type error: the first line is about vectors, the second line is about numbers).

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You'll just get the negative if you do it that way, which gives the same answer for the area.

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