2.2 Coordinate systems and components of a vector  (Page 3/15)

When we know the scalar components $A_x$ and $A_y$ of a vector $\overrightarrow{A}$ , we can find its magnitude A and its direction angle ${\theta }_A$ . The direction angle    —or direction, for short—is the angle the vector forms with the positive direction on the x -axis. The angle ${\theta }_A$ is measured in the counterclockwise direction from the + x -axis to the vector ( [link] ). Because the lengths A , $A_x$ , and $A_y$ form a right triangle, they are related by the Pythagorean theorem:

This equation works even if the scalar components of a vector are negative. The direction angle ${\theta }_A$ of a vector is defined via the tangent function of angle ${\theta }_A$ in the triangle shown in [link] :

When the vector lies either in the first quadrant or in the fourth quadrant, where component $A_x$ is positive ( [link] ), the angle $\theta$ in [link] ) is identical to the direction angle ${\theta }_A$ . For vectors in the fourth quadrant, angle $\theta$ is negative, which means that for these vectors, direction angle ${\theta }_A$ is measured clockwise from the positive x -axis. Similarly, for vectors in the second quadrant, angle $\theta$ is negative. When the vector lies in either the second or third quadrant, where component $A_x$ is negative, the direction angle is ${\theta }_A=\theta +180\text{°}$ ( [link] ).

In many applications, the magnitudes and directions of vector quantities are known and we need to find the resultant of many vectors. For example, imagine 400 cars moving on the Golden Gate Bridge in San Francisco in a strong wind. Each car gives the bridge a different push in various directions and we would like to know how big the resultant push can possibly be. We have already gained some experience with the geometric construction of vector sums, so we know the task of finding the resultant by drawing the vectors and measuring their lengths and angles may become intractable pretty quickly, leading to huge errors. Worries like this do not appear when we use analytical methods. The very first step in an analytical approach is to find vector components when the direction and magnitude of a vector are known.