The direction of a vector in a coordinate system is defined by the angle it forms with the positive x-axis.

In a two-dimensional coordinate system, the direction of a vector is typically described by the angle it makes with the positive x-axis. This angle is measured in a counterclockwise direction and is conventionally represented by the Greek letter $\theta$. If the components of the vector along the x and y axes are known, the direction can be calculated using trigonometric functions.

For example, if a vector has components $(x, y)$, the angle $\theta$ can be determined using the tangent function:

To find the angle $\theta$, you can use the inverse tangent function, also known as $\arctan$ or $\tan^{-1}$. It is important to note that this function yields an angle in the range of $-90^\circ$ to $+90^\circ$. Therefore, you may need to adjust the angle by adding or subtracting $180^\circ$ to obtain the correct direction, depending on the quadrant in which the vector is located.

In a three-dimensional coordinate system, describing the direction of a vector becomes more intricate, as it involves two angles. One effective way to specify the direction is by using spherical coordinates. In this system, the direction is represented by two angles: the azimuthal angle $\phi$, which is the angle in the xy-plane from the positive x-axis, and the polar angle $\theta$, which is the angle from the positive z-axis.

To find these angles, consider a vector with components $(x, y, z)$. The azimuthal angle $\phi$ can be calculated similarly to the 2D case:

The polar angle $\theta$ can be determined using the cosine function:

where $r$ is the magnitude of the vector, calculated using the Pythagorean theorem:

Once you have $r$, you can find the angle $\theta$ using the inverse cosine function, also known as $\arccos$ or $\cos^{-1}$.

Remember that the direction of a vector is always relative to the coordinate system being used. Therefore, it is essential to clearly define your coordinate system when describing the direction of a vector.