How do you determine the direction of a vector in a coordinate system?

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

In a two-dimensional coordinate system, the direction of a vector is usually described by the angle it makes with the positive x-axis. This angle is measured in a counterclockwise direction and is often represented by the Greek letter theta (θ). The direction can be calculated using trigonometric functions if the components of the vector along the x and y axes are known.

For instance, if a vector has components (x, y), the direction θ can be found using the tangent function: tan(θ) = y/x. The angle θ can then be found using the inverse tangent function, also known as arctan or tan^-1. It's important to note that this will give an angle in the range -90° to +90°, so you may need to add or subtract 180° to get the correct direction depending on the quadrant in which the vector lies.

In a three-dimensional coordinate system, the direction of a vector is a bit more complex to describe as it involves two angles. One way to specify the direction is by using spherical coordinates, where the direction is given by two angles: the azimuthal angle φ, which is the angle in the xy-plane from the positive x-axis, and the polar angle θ, which is the angle from the positive z-axis.

To find these angles, if a vector has components (x, y, z), the azimuthal angle φ can be found similarly to the 2D case: φ = arctan(y/x). The polar angle θ can be found using the cosine function: cos(θ) = z/r, where r is the magnitude of the vector, which can be found using the Pythagorean theorem: r = sqrt(x^2 + y^2 + z^2). The angle θ can then be found using the inverse cosine function, also known as arccos or cos^-1.

Remember, the direction of a vector is always relative to the coordinate system you're using, so it's important to clearly define your coordinate system when describing the direction of a vector.