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Hello, I thought I understood the Dot Product but Apparently Not!

[tex] \overline{u} \ \cdot \ \overline{v} \ = (u_x \ \cdot \ v_x) ( \overline{i} \cdot \overline{i} ) \ + \ (u_y \ \cdot \ v_y) ( \overline{j} \cdot \overline{j} ) \ = \ | \overline{u} | | \overline{v} | cos \theta [/tex]

That is the dot product in all it's glory, right? I've included the component definition & the angle definition.

This is all I've ever needed in physics anyway.

I guess I took a lot of stuff for granted about vectors & this book is ironing out the mental kinks,

http://img121.imageshack.us/img121/6545/vector1n.jpg [Broken]

I believe the above way is the method for adding two vectors together, is in not?

What has confused me about this is that in my book there are plenty of pictures of the following;

http://img510.imageshack.us/img510/2930/vector2.jpg [Broken]

And a picture such as the above is used to define the Inner Product.

I thought adding vectors tail to tail was wrong, it's has to be like in my first picture, (as my physics book made quite clear)?

Which way is which and when is either done???

[tex] \overline{u} \ \cdot \ \overline{v} \ = (u_x \ \cdot \ v_x) ( \overline{i} \cdot \overline{i} ) \ + \ (u_y \ \cdot \ v_y) ( \overline{j} \cdot \overline{j} ) \ = \ | \overline{u} | | \overline{v} | cos \theta [/tex]

That is the dot product in all it's glory, right? I've included the component definition & the angle definition.

This is all I've ever needed in physics anyway.

**However!**My wonderful book from the 1960's has so kindly informed me of the definition of the dot product arises from the Law of Cosines. I'm at a loss I must say.I guess I took a lot of stuff for granted about vectors & this book is ironing out the mental kinks,

**bear with me if I say something stupid, it just has to be done****I need to get a few things clear first, should only take a moment!****1.****Adding Vectors!**http://img121.imageshack.us/img121/6545/vector1n.jpg [Broken]

I believe the above way is the method for adding two vectors together, is in not?

**(The absolute value signs on [tex] \overline{u} \ + \ \overline{v}[/tex] is a mistake, so is the labelling on the graph, vector u is at point 2, NOT 1, I apologise!).**What has confused me about this is that in my book there are plenty of pictures of the following;

http://img510.imageshack.us/img510/2930/vector2.jpg [Broken]

And a picture such as the above is used to define the Inner Product.

I thought adding vectors tail to tail was wrong, it's has to be like in my first picture, (as my physics book made quite clear)?

Which way is which and when is either done???

*I don't mean to take up too much of anybodies time with my misunderstandings but I'll have a few more questions*
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