2.7: Algèbre linéaire : méthode des composantes
It is cumbersome to find the magnitudes of vectors with the use of the parallelogram rule or the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as is done in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
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 on a strong windy day. 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 first step in an analytical approach is to find vector components when the direction and magnitude of a vector are known. Resolving vectors into their scalar components (i.e., finding their scalar components) and expressing them analytically in vector component form allows us to use vector algebra to find sums or differences of many vectors analytically (i.e., without using graphical methods). For example, to find the resultant of two vectors, we simply add them component by component. Analytical methods for finding the resultant and, in general, for solving vector equations are very important in physics because many physical quantities are vectors. For example, we use this method in kinematics to find resultant displacement vectors and resultant velocity vectors, in mechanics to find resultant force vectors and the resultants of many derived vector quantities, and in electricity and magnetism to find resultant electric or magnetic vector fields.
Multiplication of a vector by a scalar gives a vector quantity and is known as scalar multiplication. When a vector is multiplied by a positive scalar, the result is a new vector that is parallel to the given. The magnitude of this new vector is obtained by multiplying the magnitude of the original vector by a scalar.
This text is adapted from Openstax, University Physics Volume 1, Section 2.3: Algebra of Vectors.
