In maths, we use the symbols \(\times\) and \(\cdot\) for the same purpose. For example, \[\displaystyle \color{Blue}{\left| 5\times 4\right| =\left| 5\cdot 4\right| }\quad \quad \quad\text{as each of these is} = 20\]

But if you do that in vectors, like \[\left| \vec{a}\times \vec{b} \right|=\left|\vec{a} \cdot \vec{b} \right|\] then it's a \(\color{Red}{\textbf{Big Mistake !!!}}\)

But for some integer valued magnitudes of \(\vec{a}\) and \(\vec{b}\) in the range \(0\leq a,b \leq 10\) and for angle \(\theta\) (with some integer value in degrees, \(\color{Purple}{0^\circ \leq \theta \leq 180^\circ}\)) between them, the above said \(\color{Red}{\text{false}}\) property is observed to be \(\color{Green}{\text{true}}\).

Find the number of ordered triples (\(\color{Blue}{\mathrm{a,b,\theta}}\)).

**Details and assumptions** :-

\(\bullet \) \(a\) and \(b\) are the magnitudes of \(\vec{a}\) and \(\vec{b}\) respectively.

\(\bullet\) For an angle \(\theta\) between vectors \(\vec{a}\) and \(\overline{b}\) (Of course it is considered as the angle in range \(0\leq \theta \leq 180^\circ\) ), it's defined as \(\left|\vec{a}\cdot \vec{b}\right| =\left|a b \cos{\theta} \right|\) and \(\left|\vec{a}\times \vec{b}\right| =\left|ab\sin\theta\right|\).

This problem is a part of the set Mistakes Give Rise to Problems!!!

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