geometry - What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors?

I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean?

More specifically,

• Why is it that dot product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\cos\theta$?


• Why is it that cross product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\sin\theta$, times a unit vector determined from the right-hand rule?

To me, both these formulae seem to be arbitrarily defined (although, I know that it definitely wouldn't be the case).

If the cross product could be defined arbitrarily, why can't we define division of vectors? What's wrong with that? Why can't vectors be divided?

Answer

I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean?

Perhaps you would find the geometric interpretations of the dot and cross products more intuitive:

The dot product of A and B is the length of the projection of A onto B multiplied by the length of B (or the other way around--it's commutative).

The magnitude of the cross product is the area of the parallelogram with two sides A and B. The orientation of the cross product is orthogonal to the plane containing this parallelogram.

Why can't vectors be divided?

How would you define the inverse of a vector such that $\mathbf{v} \times \mathbf{v}^{-1} = \mathbf{1}$? What would be the "identity vector" $\mathbf{1}$?

In fact, the answer is sometimes you can. In particular, in two dimensions, you can make a correspondence between vectors and complex numbers, where the real and imaginary parts of the complex number give the (x,y) coordinates of the vector. Division is well-defined for the complex numbers.

The cross-product only exists in 3D.

Division is defined in some higher-dimensional spaces too (such as the quaternions), but only if you give up commutativity and/or associativity.

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Here's an illustration of the geometric meanings of dot and cross product, from the wikipedia article for dot product and wikipedia article for cross product: