Dot Products -- Some questions to help my understanding

[JackFyre]

TL;DR

Just a little confusion regarding cross products.

Just a little doubt regarding vector dot products-
From what I understand(which may be wrong), dot products are the products of the magnitudes of the two vectors.
The equation is given as a · b= |a|x|b|cos θ
Can this be understood as the magnitude of a x the magnitude of b in the direction of vector a?
if so, don't dot products inherently have a direction?

So in this picture, won't the direction of the product be in the direction of vector B?
(I know I am mistaken, just need a clarification that will clear up my confusion!)

Cheers!

 

[PeroK]

Summary:: Just a little confusion regarding cross products.

Just a little doubt regarding vector dot products-
From what I understand(which may be wrong), dot products are the products of the magnitudes of the two vectors.
The equation is given as a · b= |a|x|b|cos θ
Can this be understood as the magnitude of a x the magnitude of b in the direction of vector a?
if so, don't dot products inherently have a direction?
View attachment 275496So in this picture, won't the direction of the product be in the direction of vector B?
(I know I am mistaken, just need a clarification that will clear up my confusion!)

Cheers!

You are talking about the dot or scalar product. Not to be confused with the cross or vector product.

The dot product itself is a scalar by definition: $$\vec a \cdot \vec b = |\vec a||\vec b| \cos \theta$$
It is, however, sometimes useful to consider the vector: $$(\vec a \cdot \vec b) \hat a \ \ \text{or} \ \ (\vec a \cdot \vec b) \hat b $$

 

[berkeman]

It is, however, sometimes useful to consider the vector:

Interesting. What is that used for?

 

[PeroK]

Interesting. What is that used for?

The Gram-Schmidt orthogonalisation process, for example:

https://en.wikipedia.org/wiki/Gram–Schmidt_process

 

[kuruman]

Summary:: Just a little confusion regarding cross products.

if so, don't dot products inherently have a direction?

No. Direction is defined relative to chosen coordinate axes, e.g. "east", "north", $\hat x$, $\hat y$, etc. The projection of $\vec A$ onto $\vec B$ will be the same number regardless of the orientation of the two vectors relative to the coordinate axes used to describe them.

The component of $\vec A$ in the direction of $\vec B$ can be written as a dot product $A_B=\vec A \cdot \left(\dfrac{\vec B}{B}\right)=\vec A\cdot\hat B$. Note that the fraction in parentheses is a unit vector in the direction of $\vec B$ and that if you replace "$B$" with "$x$", you get the familiar form $A_x=\vec A\cdot\hat x.$ In short, $A_B$ has as much "inherent" direction as $A_x$.

 

[A.T.]

Just a little confusion regarding cross products.
Just a little doubt regarding vector dot products

Here are good explanations on both: