Scalar Product

## Question:

I was just wondering why the DOT product is considered to be a
scalar. If I take and DOT A with B, then you are saying that we
are multiplying the component of vector A along B, directed in
the same direction as B, with the Magnitude of Vector B. Now we
know that the Product will be in the same direction as B directed
along it's unit vector (along it's line of sight);
therefore, why do we not assign the product the same unit vector
as B??
Thanks for the assistance!

## Answer:

We can calculate the dot product by taking two numbers and
multiplying them together. One number is the length of one of the
vectors. That is a scalar quantity. The other number is the
length of the other vector, a scalar, times the cosine of the
angle between them, a scalar, so our second factor, being the
product of two scalars is a scalar. Now when we multiply our two
factors, both scalar, together the result is the dot, or scalar,
product.
It happens that the second factor in our dot product is the
length of one vector projected onto the other, but that
projection is just a number, not a vector. The scalar nature of
the dot product can be confirmed by carrying out the term by term
multiplication of the two vectors, just as though they were
polynomials, including in each term the unit vector which gives
the component its direction. The dot product of two different
unit vectors is 0. The dot product of a unit vector with itself
is 1. Applying these rules leaves us with a scalar when the
multiplication is complete.

This information is brought to you by M. Casco Associates, a
company dedicated to helping humankind reach the stars through
understanding how the universe works. My name is James D. Jones.
If I can be of more help, please let me know.

JDJ