Why does x1i =r1 -y1j in Lorentz electromagnetic force?

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The discussion centers on the Lorentz electromagnetic force and a specific notation in a physics text. The user expresses confusion regarding the equation x1i = r1 - y1j, questioning its derivation and the use of the Pythagorean theorem. Clarification is provided that the equation refers to the vector representation of r, defined as r = xi + yj, and emphasizes that the magnitude of r is calculated using the Pythagorean theorem, resulting in |r| = √(x² + y²). The distinction between the vector r and its magnitude is highlighted as a key point of understanding. Additionally, a follow-up question is raised about the equation vy1k = v x r on page 6, indicating further inquiry into the relationships between velocity and position vectors in the context of the discussion.
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When going over my notes on the lorentz electromagnetic force in my physics 3 electromagnetism class got a bit puzzled by one statement that seems just to be taken for granted. Could anyone help me and tell me why x1i =r1 -y1j on page 5 on http://www.physics.gla.ac.uk/~dland/ELMAG305/Elmag305txt1.pdf where frame 1 is the stationary frame and 2 the frame moving with velocity v to the right of frame 1

If its just by using pythagorus wouldn't the above quantities (x,y and r) be to the second power. i just can't get my head around this and realize its probably something really straight forward. any help would be much appreciated.
 
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the vector r = xi + yj by definition, see the figure on the previous page.

Using the Pythograean theorem you get the magnetude of r.

\mid \hat {r} \mid = \sqrt { x^2 + y^2 }

You are dealing with the VECTOR r. See the difference?
 
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haha yeah i do now and realize how obvious that was, thanks
 
Another quick question, why on page 6 of the slide does vy1k = v x r?
 

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