I Radial Vector in Cartesian form

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If I wanted to write ##\hat{r}##in terms of ##\hat{x}##and ##\hat{y}##, is it ##\frac{\hat{x} + \hat{y}}{\sqrt{2}}## ?
 

tnich

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If I wanted to write ##\hat{r}##in terms of ##\hat{x}##and ##\hat{y}##, is it ##\frac{\hat{x} + \hat{y}}{\sqrt{2}}## ?
Only if ##x=y\ge 0##.
In general ##\hat r= \frac{x\hat x + y\hat y} {\sqrt {x^2+y^2}}## for ##\sqrt {x^2+y^2}>0##.
 

PeroK

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Only if ##x=y\ge 0##.
In general ##\hat r= \frac{x\hat x + y\hat y} {\sqrt {x^2+y^2}}## for ##\sqrt {x^2+y^2}>0##.
... as

1) ##\vec r= x\hat x + y\hat y##

2) ##\hat r = \frac{\vec r}{|\vec r|}##

3) ##|\vec r| = \sqrt{x^2 + y^2}##
 

Stephen Tashi

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If I wanted to write ##\hat{r}##in terms of ##\hat{x}##and ##\hat{y}##, is it ##\frac{\hat{x} + \hat{y}}{\sqrt{2}}## ?
What does the "##\hat{}##" in your notation signify? Is it only to indicate that variables are vectors? - or does it indicate vectors of length 1?

What is your definition of ##\hat{r}##?
 
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I found an E field in the form of ##\vec{E} = C(\frac{1} {|\vec{r}|} - \frac{1} {|\vec{r} - \vec{d}|})\hat{r}## where C is a constant.

I need to transform this into x,y coordinates. So I wrote

##\vec{E} = C(\frac{1} {\sqrt{x^2 + y^2}} - \frac{1} {\sqrt{(x-d)^2 + (y)^2)}}) \frac{x\hat{i} + y\hat{j}}{\sqrt{x^2 + y^2}}##
 
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PeroK

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I found an E field in the form of ##\vec{E} = C(\frac{1} {|\vec{r}|} - \frac{1} {|\vec{r} - \vec{d}|})\hat{r}## where C is a constant.

I need to transform this into x,y coordinates. So I wrote

##\vec{E} = C(\frac{1} {\sqrt{x^2 + y^2}} - \frac{1} {\sqrt{(x-d)^2 + (y-d)^2)}}) \frac{x\hat{i} + y\hat{j}}{\sqrt{x^2 + y^2}}##
It looks like you have ##\vec d = (d, d)## there.
 

PeroK

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What does the "##\hat{}##" in your notation signify? Is it only to indicate that variables are vectors? - or does it indicate vectors of length 1?

What is your definition of ##\hat{r}##?
##\hat r## is a unit vector.
 

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