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By the end of this section, you will be able to:
  • Describe vectors in two and three dimensions in terms of their components, using unit vectors along the axes.
  • Distinguish between the vector components of a vector and the scalar components of a vector.
  • Explain how the magnitude of a vector is defined in terms of the components of a vector.
  • Identify the direction angle of a vector in a plane.
  • Explain the connection between polar coordinates and Cartesian coordinates in a plane.

Vectors are usually described in terms of their components in a coordinate system . Even in everyday life we naturally invoke the concept of orthogonal projections in a rectangular coordinate system. For example, if you ask someone for directions to a particular location, you will more likely be told to go 40 km east and 30 km north than 50 km in the direction 37 ° north of east.

In a rectangular (Cartesian) xy -coordinate system in a plane, a point in a plane is described by a pair of coordinates ( x , y ). In a similar fashion, a vector A in a plane is described by a pair of its vector coordinates. The x -coordinate of vector A is called its x -component and the y -coordinate of vector A is called its y -component. The vector x -component is a vector denoted by A x . The vector y -component is a vector denoted by A y . In the Cartesian system, the x and y vector components    of a vector are the orthogonal projections of this vector onto the x - and y -axes, respectively. In this way, following the parallelogram rule for vector addition, each vector on a Cartesian plane can be expressed as the vector sum of its vector components:

A = A x + A y .

As illustrated in [link] , vector A is the diagonal of the rectangle where the x -component A x is the side parallel to the x -axis and the y -component A y is the side parallel to the y -axis. Vector component A x is orthogonal to vector component A y .

Vector A is shown in the x y coordinate system and extends from point b at A’s tail to point e and its head. Vector A points up and to the right. Unit vectors I hat and j hat are small vectors pointing in the x and y directions, respectively, and are at right angles to each other. The x component of vector A is a vector pointing horizontally from the point b to a point directly below point e at the tip of vector A. On the x axis, we see that the vector A sub x extends from x sub b to x sub e and is equal to magnitude A sub x times I hat. The magnitude A sub x equals x sub e minus x sub b. The y component of vector A is a vector pointing vertically from point b to a point directly to the left of point e at the tip of vector A. On the y axis, we see that the vector A sub y extends from y sub b to y sub e and is equal to magnitude A sub y times j hat. The magnitude A sub y equals y sub e minus y sub b.
Vector A in a plane in the Cartesian coordinate system is the vector sum of its vector x - and y -components. The x -vector component A x is the orthogonal projection of vector A onto the x -axis. The y -vector component A y is the orthogonal projection of vector A onto the y -axis. The numbers A x and A y that multiply the unit vectors are the scalar components of the vector.

It is customary to denote the positive direction on the x -axis by the unit vector i ^ and the positive direction on the y -axis by the unit vector j ^ . Unit vectors of the axes , i ^ and j ^ , define two orthogonal directions in the plane. As shown in [link] , the x - and y - components of a vector can now be written in terms of the unit vectors of the axes:

{ A x = A x i ^ A y = A y j ^ .

The vectors A x and A y defined by [link] are the vector components of vector A . The numbers A x and A y that define the vector components in [link] are the scalar component     s of vector A . Combining [link] with [link] , we obtain the component form of a vector :

A = A x i ^ + A y j ^ .

If we know the coordinates b ( x b , y b ) of the origin point of a vector (where b stands for “beginning”) and the coordinates e ( x e , y e ) of the end point of a vector (where e stands for “end”), we can obtain the scalar components of a vector simply by subtracting the origin point coordinates from the end point coordinates:

Practice Key Terms 8

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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