For two vectors $2i + 3j - 4k$ and $i + 2k$. The way I understood crops product is that we can find these two vectors magnitude. Let us say (Assumption ) $= 10$ and $7$. Now , to find area . We do $\frac 12 \times 10 \times 7 \times \sin \theta$. Probably because $10$ and $7$ are parallel to each other. And if we draw something that is not parallel to $10$ units. Then , we can find an area. This is one way.
Second way is that we write it in discriminant form.
Now , my confusion is that solving it in discriminant form , It gives area. Now , how did we take care of sin theta here and why do we write it in discriminant form.
Just a picture of how I view these Values.
Area of a triangle formed by two adjacent vectors $\vec A$ and $\vec B$ is given by $\frac{1}{2} |\vec A \times \vec B|= \frac{1}{2} |\vec A| |\vec B| \sin \theta$. If you find area by LHS in above $\sin \theta $ as taken care of. Other thing is the cros product is represented by the determinant which gives you a vector, them find its modulus.