How to Compute the Magnitudes and Directions of Vectors
Vectors are graphical representations of quantities that have both size and directions. This special mathematical language, used in many physics, engineering and navigational fields, is necessary when describing the behavior of physical quantities such as displacement, velocity and acceleration. Vector quantities themselves can be described by their components. A vector component is the value of the vector projected onto an axis. Like all geometry problems, you must have certain measurements to solve for others. In the case of vectors, the vector components must be known to solve for an unknown magnitude and to determine its direction.
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Instructions
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Convert the known component notation into magnitude-angle notation. For example, if you know the component a - x to be 7 and the component a - y to be 5, the magnitude of the vector A can be determined using the Pythagorean theorem. In this case, the equation becomes: √((a - x)^2 + (a - y)^2) = A = √(7^2 + 5^2) = sqrt(74) = 8.6. The magnitude of A is 8.
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Determine the angle T that the vector forms with the x-axis. This is determined with the equation tan(T) = (a - y / a - x). Solving for T, the equation becomes: T = arctan(a - y / a - x) = arctan(5 / 7) = 35.5 degrees. Therefore, the vector A is inclined 35.5 degrees, counterclockwise, from the x-axis.
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Determine the direction of the vector by examining its counterclockwise angle from the x-axis. For example, vector A is inclined at 35.5 degrees from the x-axis, which is moving in the positive direction, to the northeast.
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