Finding Vector Components of A and B in Spherical Coordinates

[Tom McCurdy]

Homework Statement

I have a vector A and B in spherical coordinates, and I need to find:

• part a) The vector component of B in the direction of A.
• part b) The vector component of B perpendicular to A

Homework Equations

dot product
cross product

The Attempt at a Solution

Alright at my first look I saw vector component and I thought of the cross product which yields a vector result, but then I realized that it in no way tells us anything about B in the direction of A. So then dot product came to mind but that yeilds a scaler result. So I am kind of stumped as to where to start.

My only idea for part a was to take (A dot B)/vector A which would yield a vector result.
and I had no idea what to do for part b.

 

[robphy]

What can you say about:
(vector B) minus (your answer to part a) ?

 

[Tom McCurdy]

wouldn't you just get vector B pointing the other way

i am not sure what you are hinting at

 

[robphy]

Draw a picture... and maybe form dot and/or cross products with "(vector B) minus (your answer to part a)".

 

[Tom McCurdy]

the thing is I still don't have an answer to a, so its hard for me to know what you mean

 

[Gokul43201]

A dot B gives you a scalar - and this is the magnitude of the answer that you want in (a). But the direction is simply that of vector A. So, you can construct the vector component by simply multiplying this scalar by the unit vector along A.

As suggested by robphy, draw a diagram...everything will fall into place.

 

[robphy]

the thing is I still don't have an answer to a, so its hard for me to know what you mean

I thought you had part a... or something close to it.
To make life easier, work with the unit vector along $$\vec A$$, namely $$\hat A = \frac{1}{A}\vec A$$.

As a simpler example, consider $$\hat A=\hat x$$.
How would you find the vector component of $$\vec B$$ along the x-direction?
What can be said about the "remaining piece" in terms of the x-direction?