# Mistakes give rise to Problems- 17

Algebra Level 5

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

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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