# Why is the $\theta$ in cross product defined as the angle between the tails of $\vec{r}$ and $\vec{F}$?

I have a textbook that tells me that the definition of theta in cross product is the angle between the tails of r and F. However, they proceed to solve many book problems head to tail.

Cross product returns the vector perpendicular to the two vectors that form it. For example, if a parallelogram (rectangle) is made by i and j, vector k is perpendicular to both i and j (or orthogonal to the ij plane).

Here's the deal: because sine is positive in the first quadrant and the second quadrant, theta or 180-theta give you the same answer when used in cross product.

Why does the book bother indicating that the theta must be defined as tail to tail when head to tail gets you the same answer? What would be the theoretical or practical reasons that cross product theta "must" be defined this way, and if it must be defined this way, why does solving head to tail get you the right answer?

The cross product physically means the area element of the parallelogram formed by two vectors. You can see this parallelogram when you put the vector tails together, but you can't see it when you put them head-to-tail. However, the head-to-tail picture is better for visualizing vector addition, which is why your textbook uses it in other places.

Once you start a second physics course, I promise nobody will ever care about this distinction again. I would be very surprised if it was ever emphasized at all, even in a first course.