How Do Squared Vector Components Relate to the Dot Product?

AI Thread Summary
Squared vector components relate to the dot product through the relationship between vector magnitudes and their components in a Cartesian system. The magnitude of a vector, represented as r, is calculated using the square root of the sum of the squares of its components, expressed as r = sqrt(r1^2 + r2^2 + r3^2). This leads to the equation r^2 = r1^2 + r2^2 + r3^2, demonstrating that the dot product of a vector with itself, denoted as r·r, equals r^2 when the angle θ between the vectors is zero. The discussion emphasizes understanding how vector components project onto axes and the application of the Pythagorean theorem in this context. Overall, the relationship between squared components and the dot product is fundamental in vector analysis.
Oblio
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This technically a homework question, but needed for homework and understanding for homework to come. Just hope to get it cleared up, thanks again!

1. This formula is given:

r_{1}s_{1} + r_{2}s_{s}+ r_{e}s^{3} = \sumr_{n}s_{n} (with the limits etc. not too important).

Then, in respect to scalar products, the magnitude of any vector is denoted by l r l or by Pythoagora's theorem: square root of[r(1)^{2} + r(2)^{2} + r(3)^{2}]
(couldn't find square root in latex)

and that THAT is the same as square root of [r . r]
This last step I do not follow...
 
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I know that it's true, (in reading my text) but how does 3 squared terms boil down to r^2?
 
This is a vector operation.

One squares the corresponding vector components represented in each of three dimensions, which are orthogonal in the Cartesian system.

The magnitude r is given as the sqrt of the sum of the squares, i.e. r = sqrt (r12 + r22 + r32), so

r2 = r12 + r22 + r32
 
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Oblio said:
I know that it's true, (in reading my text) but how does 3 squared terms boil down to r^2?

See \vec{r}\cdot\vec{r} is nothing but \sca{r}\sca{r}\cos\theta. But \theta = 0 and so \cos\theta = 1.

Hence \vec{r}\cdot\vec{r} = ^{}r^2

Now a projection of a vector on an axis is known as the component of the vector on that axis. So take a cartesian system and draw any arbritary vector and drop perpendiculars on the x y and z axis. Now try using pythagoras theorem for this and you will see that how

\sca{r^2} = \sqrt{r^2_x + r^2_y + r^2_z}
 
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