Question

Determine the direction angle counterclockwise from the x-axis.

Answer 1

The direction angle of a vector is the angle measured counterclockwise from the positive x-axis. To get this angle, use trigonometric functions or physical measurements with a protractor, remembering that counterclockwise angles are positive and clockwise angles are negative.

Step-by-step explanation:

The direction angle of a vector is measured counterclockwise from the positive direction of the x-axis. To determine this direction angle, you can use trigonometric functions or a protractor to measure the angle counterclockwise from the positive x-axis to the vector in question. The angles in the counterclockwise direction are considered positive, according to conventional mathematics rules.

When dealing with vectors in different quadrants, the direction angle is treated differently. For example, if a vector lies in the first or fourth quadrant, its direction angle A is positive when the x-component (Ax) of the vector is positive. In contrast, for vectors in the second or third quadrant where Ax is negative, the direction angle A is calculated as θA = θ + 180°. Remember, the clockwise direction is assigned a negative angle.

Magnetic fields and current directions also adhere to the counterclockwise rule, as exemplified by the right-hand rule, where current direction dictates that the fingers curl in the counterclockwise direction around the conductor.

Answer 2

To determine the direction angle, measure counterclockwise from the positive x-axis to the vector. Positive angles are measured counterclockwise, while negative angles are clockwise. In the second and third quadrants, add 180° to the measured angle for the correct direction angle.

Step-by-step explanation:

To determine the direction angle counterclockwise from the x-axis, one must understand that angles in a coordinate system are measured starting from the positive direction of the x-axis. The angle measured in this manner is the angle formed by the positive x-axis and the vector, moving counterclockwise towards the vector's direction. This is also known as the global or standard position of an angle. Angles measured counterclockwise are considered positive, while angles measured clockwise are negative.

When a vector is in the first or fourth quadrant, the direction angle is exactly the angle we measure from the positive x-axis to the vector. If the vector falls in the second or third quadrant (where the x-component of the vector is negative), the direction angle is obtained by adding 180° to the angle measured from the positive x-axis. This adjustment is necessary to reflect the fact that we are moving counterclockwise from the positive x-axis to point in the direction of the vector. For example, a vector pointing south of west would have a global angle of 210°.

For practical purposes, when sketching vectors or using a protractor, this angle can be determined visually. However, in most calculations, trigonometric relationships are employed to find this angle without the need for a protractor.

It is essential to observe the sign of the vector components, which can indicate the quadrant in which the vector lies and consequently influence the calculation of the direction angle. If the vector components include a negative value, this suggests that the direction angle will differ from the angle calculated directly with trigonometric relationships, typically requiring additional consideration to find the proper direction angle.