The analytical method of vector addition can be described as a combination of steps that utilize the Pythagorean theorem as well as trigonometric identities. These two concepts determine magnitude and direction of the resultant vector.
To sum vectors using the analytical method, first determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of the vectors using equations that involve the sine and cosine of the angles formed by the vectors. Those equations are Ax = Acos(theta), Bx = Bcos(theta), Ay = Asin(theta) and By = Bsin(theta). Add the horizontal and vertical components of each vector to determine components Rx and Ry and the resultant vector R. To do so, use the following equations: Rx = Ax + Bx and Ry = Ay + By.
Use the Pythagorean theorem to determine the magnitude R and the resultant vector. To do this, use the equation R = the square root of Rx (squared) + Ry(squared). The direction of a resultant vector is determined by use of trigonometric functions. The three functions used in vector addition are the sine, cosine and tangent. All three functions show relationships between the measure of the angle to the ratio of the lengths of the sides that are either opposite or adjacent to the angle. The angle of interest in an acute angle is a right triangle.