\(\textit Have\quad you\quad ever\quad exploited\quad symbolic\quad notation\quad of\quad Mathematics?\)

Can you prove ?

\(\Huge\sqrt { { x }^{ 2 } } =\left| x \right|\quad \quad \quad ...(1)\)

If yes, then read further else skip this note.

Do you know what is 'Scalar Product of two vectors'?

- Scalar product of two vectors: For vectors \(\displaystyle \overrightarrow { A }\) and \(\displaystyle \overrightarrow { B }\), the scalar product is defined by \(\displaystyle \overrightarrow { A } .\overrightarrow { B } = AB\cos { \theta }\) where \(\displaystyle { \theta }\) is the angle, in radians with \(\displaystyle 0\le \theta \le \pi\) between \(\overrightarrow { A }\) and \(\displaystyle \overrightarrow { B }\). This is a scalar quantity; that is, a real number, not a vector. The scalar product has the following property:

\[\overrightarrow { A } .\overrightarrow { B } =\overrightarrow { B } .\overrightarrow { A } \\ \overrightarrow { A } .\overrightarrow { A } ={ \left| \overrightarrow { A } \right| }^{ 2 }={ A }^{ 2 }\\ \overrightarrow { A } .\left( \overrightarrow { B } +\overrightarrow { C } \right) =\overrightarrow { A } .\overrightarrow { B } +\overrightarrow { A } .\overrightarrow { C }\]

Now, the climax:

If \(\displaystyle \overrightarrow { A }\) and \(\overrightarrow { B } \) are two vectors having an angle \(\theta\) between them[ \(\theta\) being the smaller angle]

then,

The magnitude of the sum of two vectors is written as

\(\displaystyle \left| \overrightarrow { A } +\overrightarrow { B } \right| \quad \quad \quad ...(2)\)

using (1), equation (2) can be written as

\(\displaystyle \sqrt { { { \left( \overrightarrow { A } +\overrightarrow { B } \right) }^{ 2 } } } =\left| \overrightarrow { A } +\overrightarrow { B } \right|\)

or

\(\displaystyle \left| \overrightarrow { A } +\overrightarrow { B } \right| = \sqrt { { (\overrightarrow { A } +\overrightarrow { B } ) }{ (\overrightarrow { A } +\overrightarrow { B } ) } } =\sqrt { \overrightarrow { A } .\overrightarrow { A } +2\overrightarrow { A } .\overrightarrow { B } +\overrightarrow { B } .\overrightarrow { B } }\)

Therefore,

\(\displaystyle \left| \overrightarrow { A } +\overrightarrow { B } \right| =\sqrt { { A }^{ 2 }{ +2AB\cos { \theta } +B }^{ 2 } } \). TaDa! This is the expression for the magnitude of sum of two vectors. Same thing can be derived geometrically.You can find the geometric prove in any standard textbooks.

I found this when I was confused whether 2 is the coefficient of \(AB\cos { \theta }\) or not. More significant is the message that symbolic notation in Mathematics is not just to threaten students. Sometimes they help us understand things in simple shorter ways.

Have you ever exploited symbolic notation of Mathematics? Share an example.

[Try to be as elementary as possible]

Ta-Ta.

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