# Mistakes give rise to Problems- 17

Algebra Level 5

In maths, we use the symbols $\times$ and $\cdot$ for the same purpose. For example, $\displaystyle \color{#3D99F6}{\left| 5\times 4\right| =\left| 5\cdot 4\right| }\quad \quad \quad\text{as each of these is} = 20$

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 $\color{#D61F06}{\textbf{Big Mistake !!!}}$

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, $\color{#69047E}{0^\circ \leq \theta \leq 180^\circ}$) between them, the above said $\color{#D61F06}{\text{false}}$ property is observed to be $\color{#20A900}{\text{true}}$.

Find the number of ordered triples ($\color{#3D99F6}{\mathrm{a,b,\theta}}$).

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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