Question

Why is the Dot Product a Scalar?

Answer 1

Scalars:

A scalar is a physical quantity that is completely defined by its magnitude.

Dot Product:

1. A dot product is the product of two vectors by taking the component of the second vector in the direction of the first vector.
   
2. In the above diagram, there are two vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$.
3. The dot product of these vectors can be given as: $\overrightarrow{A.}\overrightarrow{B}=\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\cos \left(\theta \right)$
4. Here, $\left|\overrightarrow{\mathrm{A}}\right|$ and $\left|\overrightarrow{\mathrm{B}}\right|$ denotes the magnitude of vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ respectively. and $\mathrm{\theta }$ is the angle between the two vectors.
5. In the dot product of both vectors, we can see that only magnitude is present without any direction, which shows that result is a scalar quantity.

Hence, the Dot Product of vectors is a scalar quantity.