Question

Why is the dot Product a scalar?

Answer 1

Hint: In order to solve this question, we should know that here dot product is asked with the context of vectors and the dot product is one of the types of product between vectors, here we will discuss about dot product.

Complete answer:
As we know, a vector represents that quantity with a definite magnitude and a direction associated with it while scalars are the quantities with only magnitude. In vector algebra other than addition and subtraction, multiplication operation is also done and Dot product is one of the types of multiplication between vectors.

Dot product between two vectors gives the scalar values because of its nature of operation defined, dot product is defined mathematically as
$\vec a.\vec b = \left| {\vec a} \right|\left| {\vec b} \right|\cos \theta $
where, $\left| {\vec a} \right|,\left| {\vec b} \right|$ are the magnitudes of the vectors a and b which will have scalar value and $\theta $ is the angle between vectors a and b, so the final value of dot product between vectors a and b will have a scalar value.

For example: if
$
  \vec a = 2\hat i + 3\hat j \\
  \vec b = 3\hat i + 2\hat j \\
 $

and angle between them is $\theta = {60^0}$ then dot product between the vectors a and b will be
$
  \vec a.\vec b = \left| {\vec a} \right|\left| {\vec b} \right|\cos \theta \\
  \vec a.\vec b = \sqrt {13} .\sqrt {13} .\dfrac{1}{2} \\
  \vec a.\vec b = 6.5 \\
 $
so, we see that dot product of two vectors is scalar quantity.

Hence, Due to specific nature of dot product operation as $\vec a.\vec b = \left| {\vec a} \right|\left| {\vec b} \right|\cos \theta $, Dot product is scalar.

Note: It should be remembered that, physically and graphically, the dot product between two vectors represents the area enclosed between them if two vectors represent the adjacent side of the parallelogram; hence, the units of the dot product will be unit square.