Why is there a minus sign in the vector sum of two perpendicular vectors?

[_Andreas]

Homework Statement

The vector $$\vec{E}_n$$ is the vector sum of the two vectors $$\vec{E}_r$$ and $$\vec{E}_{\theta}$$, which are perpendicular to each other (see attached picture). Calculate the magnitude of $$\vec{E}_n$$.

The Attempt at a Solution

$$E_n=E_r\cos(\theta)+E_{\theta}\sin(\theta)$$

But this is wrong. It's supposed to be a minus sign instead of a plus. I have no idea why. I thought it would be to simply add the components of $$\vec{E}_r$$ and $$\vec{E}_{\theta}$$ along the direction of $$\vec{E}_n$$. Why the minus sign?

 

[Mark44]

Homework Statement

The vector $$\vec{E}_n$$ is the vector sum of the two vectors $$\vec{E}_r$$ and $$\vec{E}_{\theta}$$, which are perpendicular to each other (see attached picture). Calculate the magnitude of $$\vec{E}_n$$.

The Attempt at a Solution

$$E_n=E_r\cos(\theta)+E_{\theta}\sin(\theta)$$

The attachment is still pending approval, so I can't see the image you attached. Based on your description, I have a right triangle with E_n as the hypotenuse and E_r and E_theta as the legs of the triangle. E_r is the base of the triangle and E_theta is the altitude.

If my interpretation of your description is correct, your equation just above is incorrect. You've omitted the vector "caps" in the equation above, so I don't know if E_n is now supposed to be a magnitude.

In any case, you have E_n = E_r + E_theta (vectors).
So |E_r| = |E_r| cos theta and |E_theta| = |E_r| sin theta.
|E_n| = sqrt{|E_r|^2 + |E_theta|^2}

But this is wrong. It's supposed to be a minus sign instead of a plus. I have no idea why. I thought it would be to simply add the components of $$\vec{E}_r$$ and $$\vec{E}_{\theta}$$ along the direction of $$\vec{E}_n$$. Why the minus sign?

Until I can see the drawing, I don't know.

 

[Doc Al]

$$E_n=E_r\cos(\theta)+E_{\theta}\sin(\theta)$$

You are assuming that $$\vec{E}_n$$ is horizontal. If true, then your equation is OK. In general, use the Pythagorean theorem to find the magnitude as Mark44 stated.

But this is wrong. It's supposed to be a minus sign instead of a plus. I have no idea why. I thought it would be to simply add the components of $$\vec{E}_r$$ and $$\vec{E}_{\theta}$$ along the direction of $$\vec{E}_n$$. Why the minus sign?

Could it be that you have the direction of $$\vec{E}_{\theta}$$ reversed?

 

[_Andreas]

You are assuming that $$\vec{E}_n$$ is horizontal. If true, then your equation is OK. In general, use the Pythagorean theorem to find the magnitude as Mark44 stated.

Yes, $$\vec{E}_n$$ is horizontal. But about the magnitude: the result seems to be a scalar (negative surface charge density divided by permittivity).

Could it be that you have the direction of $$\vec{E}_{\theta}$$ reversed?

I've drawn the picture exactly as it appears in my book, so I doubt that. However, it does indeed seem to be standard practice to draw it in the opposite direction (opposite to the direction in which it is drawn in my book, that is).