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If I consider $\mathbf{a} \times \mathbf{b} = \mathbf{c}$, as a system where the vector $\mathbf{b}$ is rotating about an axis defined by vector $\mathbf{a}$, and vector $\mathbf{c}$ shows the linear direction which vector $\mathbf{b}$ moves as it is rotating. The faster the rotation, the larger is $\mathbf{c}$.

How fast vector $\mathbf{b}$ rotates is dependent on length of vector $\mathbf{b}$, if its head is further out, it will rotate faster. Also the angle between $\mathbf{a}$ and $\mathbf{b}$, since if the angle is small it will rotate slower and if the angle is $90^\circ$ it will rotate the fastest. By why is the size of vector a important, vector $\mathbf{a}$ is just the axis of rotation, the way I think about the cross product. How does the length of the axis of rotation affect the speed of rotation.

In other words why isn't the cross product defined as only $\mathbf{a}\times \mathbf{b} = |\mathbf{b}|\sin(\theta)$ instead of $\mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin(\theta)$. Why is $|\mathbf{a}|$ important?

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  • $\begingroup$ In this context $|a|$ gives the angular velocity, i.e. the speed at which the head of a unit vector perpendicular to $a$ will move. $\endgroup$ Aug 20, 2015 at 8:52

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Because the cross product is not defined merely for the one purpose of writing rotations efficiently but has lots of other uses, and for many of these its algebraic properties are very useful. $a\times b$ as defined is an antisymmetric bilinear product; the quantity you define isn't linear in $a$. The cross product can only be interpreted the way you're interpreting it when $a$ is a unit vector. Incidentally, an elegant and efficient way to represent angular motion is to consider the angular velocity $\vec\omega$ as a vector whose length specifies the scalar angular velocity and whose direction specifies the direction of the rotation axis. There you have two of the things you described in one vector, and if you do this, $\vec\omega\times\vec b$ gives the velocity at $\vec b$ that corresponds to the angular motion described by $\vec\omega$.

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