2.7: Vector Algebra: Method of Components
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using 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 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 windy day. Every car traveling on the bridge creates air turbulence as it moves, which results in a change in the air pressure around the car. This change in air pressure produces a force, known as wind force, that is exerted on the car and the surrounding objects, including the bridge. As each car moves across the bridge, it imparts a distinct wind force that affects the bridge from a different direction, causing it to experience varying wind pressures from multiple directions. We have already gained some experience with the geometric construction of vector sums, so finding the resultant by drawing the vectors and measuring their lengths and angles may become quickly intractable, leading to huge errors. Issues like these do not appear when we use analytical methods.
The first step in an analytical approach is to find the vector components when the direction and magnitude of a vector are known. Resolving vectors into their scalar components and expressing them analytically in vector component form allows us to use vector algebra to find the sums or differences of many vectors analytically (i.e., without using graphical methods). For example, to add two vectors, we simply add them component by component. Analytical methods for finding the resultant and, in general, solving vector equations are critical 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; this is known as scalar multiplication. When a vector is multiplied by a positive scalar, the result is a new vector parallel to the given vector. The magnitude of this new vector is obtained by multiplying the individual scalar components of the original vector by the scalar. Similarly, if a vector is multiplied by a negative scalar, a vector antiparallel to the initial vector is obtained.