Direction of E-Field in 3d space

[Zaent]

Homework Statement

I am trying to find the direction of an E-Field from a given E-Field vector in 3d space.
If I have an electric field vector of (-1i, -2j, 3k), the magnitude of the field is sqrt(-1^2 + -2^2 + 3^2) = 3.74, but how do I find the direction?

Homework Equations

tan(y/x)?

The Attempt at a Solution

In 2d space I understand that the direction would be presented as an angle in relation to the positive x-axis. e.g. E-Field is 1000 N/C, 90 degrees counterclockwise from +x-axis.

I am lost as to how this is both calculated and presented in 3 dimensions, however.

 

[jedishrfu]

Think in terms of spherical coordinates which means you need two angles one relative to the z axis as an example.

 

[quantumtimeleap]

Just specifying the vector completely (not just its magnitude) automatically gives the direction of the vector. You expressed the vector in Cartesian unit vectors. That already gives its direction. We have to know exactly what the problem is asking for.

A more well-phrased question could be answered precisely. For example:

1.) Express the direction in terms of the Euler angles.
2.) Like jedishrfu said above, express the direction in terms of $\theta$ and $\phi$ in spherical polar coordinates.
3.) Express the direction as a linear combination of the three Cartesian unit vectors.

For (1.) and (2.) you can google the formulas. For (3) simply divide the vector by its norm (length).

 

[rude man]

Homework Statement

I am trying to find the direction of an E-Field from a given E-Field vector in 3d space.
If I have an electric field vector of (-1i, -2j, 3k),

You have it already!
If you want to normalize the direction vector in order to make it a unit vector, divide by your magnitude.

 

[HalfLostAndSearching]

I would like to clarify that the direction of an electric field is typically represented by a unit vector, rather than an angle. This unit vector points in the direction of the electric field, with a magnitude of 1.

To find the direction of an electric field vector in 3D space, we can use the same method as in 2D space. First, we can calculate the magnitude of the electric field vector using the Pythagorean theorem, as you have already done. Then, we can normalize the vector by dividing each component by the magnitude, resulting in a unit vector.

In your example, the unit vector would be (-1/3.74, -2/3.74, 3/3.74) or approximately (-0.27i, -0.54j, 0.81k). This unit vector represents the direction of the electric field in 3D space.

Alternatively, we can also use the direction cosines to represent the direction of an electric field vector in 3D space. The direction cosines are the ratios of the components of the electric field vector to its magnitude. In your example, the direction cosines would be (-1/3.74, -2/3.74, 3/3.74).

I hope this helps clarify the concept of direction in 3D space for an electric field vector.