Mistakes give rise to Problems- 17

Algebra Level 5

In maths, we use the symbols ×\times and \cdot for the same purpose. For example, 5×4=54as each of these is=20\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 a×b=ab\left| \vec{a}\times \vec{b} \right|=\left|\vec{a} \cdot \vec{b} \right| then it's a Big Mistake !!!\color{#D61F06}{\textbf{Big Mistake !!!}}

But for some integer valued magnitudes of a\vec{a} and b\vec{b} in the range 0a,b100\leq a,b \leq 10 and for angle θ\theta (with some integer value in degrees, 0θ180\color{#69047E}{0^\circ \leq \theta \leq 180^\circ}) between them, the above said false\color{#D61F06}{\text{false}} property is observed to be true\color{#20A900}{\text{true}}.

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

Details and assumptions :-

\bullet aa and bb are the magnitudes of a\vec{a} and b\vec{b} respectively.

\bullet For an angle θ\theta between vectors a\vec{a} and b\overline{b} (Of course it is considered as the angle in range 0θ1800\leq \theta \leq 180^\circ ), it's defined as ab=abcosθ\left|\vec{a}\cdot \vec{b}\right| =\left|a b \cos{\theta} \right| and a×b=absinθ\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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