Fundamentals of Mechanics/Units and Vectors/Vector Algebra/Addition

There are two ways to add vectors, geometrically and algebraically. Here, we will discuss the geometric method, which is useful for solving problems without using a coordinate system. The algebraic method will be discussed later. To add two vectors A and B using the geometric method, place the head of A at the tail of B and draw a vector from the tail of A to the head of B as shown in Fig. 1.3. This method is known as the triangle method. An extension to sum up more than two vectors is shown in Fig. 1.4. An alternative procedure of vector addition using the geometric method is shown in Fig. 1.5. This is known as the parallelogram method, where C is the diagonal of a parallelogram with sides A and B. To find C analytically (see Fig. 1.6), we can extend the base of the parallelogram to point F and draw a perpendicular side from point F to point G. This creates right triangle DFG; using trigonometry, we know the following three equations hold. 1.1 1.2 1.3 Substituting Eqs. 1.2 and 1.3 into Eq. 1.1 and using the trigonometric identity gives the following. 1.4 Taking the square root of both sides of Eq. 1.4 gives the magnitude of C. 1.5 The direction of C can be found as follows. 1.6 Note that only when A and B are parallel is the magnitude of their vector sum equal to . Unlike the addition of scalar quantities, which has one and only one possible value for any two given scalars, the magnitude of the sum of two vectors can vary between and . 1.4 A jogger runs from her home a distance of 0.500 km due south and then 1.00 km to the west. Find the magnitude and direction of her resultant displacement. From Fig. 1.7, we can see that the magnitude of the resultant displacement is given by the following. The direction of R is as follows.