Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
We use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. For a geometric construction of the sum of two vectors in a plane, we follow the parallelogram rule. Suppose two vectors are at arbitrary positions. Translate either one of them in parallel to the beginning of the other vector, so that after the translation, both vectors have their origins at the same point. Now, at the end of the first vector, we draw a line parallel to the second vector. At the end of the second vector, we draw a line parallel to the first vector. In this way, we obtain a parallelogram. From the origin of the two vectors, we draw a diagonal that is the resultant of the two vectors.
The other diagonal of this parallelogram is the vector difference of the two vectors. It follows from the parallelogram rule that neither the magnitude of the resultant vector, nor the magnitude of the difference vector, can be expressed as a simple sum or difference of magnitudes of the vectors. This is because the length of a diagonal cannot be expressed as a simple sum of side lengths. If we need to add three or more vectors, we repeat the parallelogram rule for the pairs of vectors until we find the resultant of all of the resultant vectors.
Drawing the resultant vector of many vectors can be generalized using the following tail-to-head geometric construction. We select any one of the vectors as the first vector, and make a parallel translation of a second vector to a position where the origin ("tail") of the second vector coincides with the end ("head") of the first vector. Then, we select a third vector and make a parallel translation of the third vector to a position where the origin of the third vector coincides with the end of the second vector. We repeat this procedure until all the vectors are in a head-to-tail arrangement. We draw the resultant vector by connecting the origin ("tail") of the first vector with the end ("head") of the last vector. The end of the resultant vector is at the end of the last vector. Because the addition of vectors is associative and commutative, we obtain the same resultant vector regardless of which vector we choose to be first, second, third, or fourth in this construction.
Scalar multiplication of a vector gives a vector quantity. Depending on the sign associated with the scalar quantity, the direction of the vector is determined. For example, if we multiply the vector quantity with a positive scalar, the new vector will be parallel to the given vector and vice versa.
This text is adapted from Openstax, University Physics Volume 1, Section 2.3: Algebra of Vectors.
