Dot Product Clarification (Kleppner & Kolenkow p.9)

[Von Neumann]

Problem:

In Kleppner's book, Introduction to Mechanics, he states

"By writing $\vec{A}$ and $\vec{B}$ as the sums of vectors along each of the coordinate axes, you can verify that $\vec{A} \cdot \vec{B} = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}$."

He suggests summing vectors, but since the sum of two vectors vectors $\vec{A}$ and $\vec{B}$ is a new vector $\vec{C}$, I don't understand how the result could be a scalar. Am I missing something?

When I was introduced to the dot product in Stewart's Calculus, he presents it as definition.

 

[barryj]

I don't get it either. What is the context in which Kleppner makes this statement?

 

[Doc Al]

He suggests summing vectors, but since the sum of two vectors vectors $\vec{A}$ and $\vec{B}$ is a new vector $\vec{C}$, I don't understand how the result could be a scalar. Am I missing something?

He's not suggesting summing vectors but representing each vector as the sum of its components times unit vectors.

Like:
A = Ax i + Ay j + Az k

Then you can take the scalar product (of A and B) and see the result more easily.

 

[CAF123]

I have the book as well. Does he not simply mean write: $$\mathbf{A} = A_x \hat{x} + A_y \hat{y} + A_z \hat{z}\,\,\,\,\text{and}\,\,\,\, \mathbf{B} = B_x \hat{x} + B_y \hat{y} + B_z \hat{z}$$ and then take dot product.

I assume by writing vectors A and B like this is what he means by 'sum of vectors along each of the coordinate axes'.

Edit: Doc Al said same thing.

 

[wifi]

The above are correct