ES-20 Vector Resolution in Three Dimensions
A vector in three dimensions may be defined in terms of the angle it makes with the vertical (y) axis and the angle a vertical plane passed through it makes with the x-axis. These angles are known as the Euler angles.

The y component, Ay, and the component in the x-y plane, Ah, may be found as shown here.

This is shown graphically.

The horizontal component, Ah, may be broken into x and z components.

This is shown graphically.

The reverse, given a vector in Cartesian components to find the magnitude and orientation in space, may also be performed. For example, consider the diagram here.

The magnitude and direction can be determined as shown here.

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