1.2: XYZ Coordinate Frame (2024)

Vectors are usually described in terms of their components in acoordinate 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)x-y 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[latex]\vec A[/latex] in a plane is described by a pair of itsvectorcoordinates. Thex-coordinate of vector[latex]\vec A[/latex]is called itsx-component and they-coordinate of vector[latex]\vec A[/latex]is called itsy-component. The vectorx-component is a vector denoted by [latex]\vec A_x[/latex]. The vectory-component is a vector denoted by [latex]\vec A_y[/latex]. In the Cartesian system, thexandyvector componentsof a vector are the orthogonal projections of this vector onto thex– andy-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:

[latex]\vec A =\vec A_x+\vec A_y[/latex]

As illustrated in the figure below, vector [latex]\vec A[/latex] is the diagonal of the rectangle where thex-component [latex]\vec A_x[/latex] is the side parallel to thex-axis and they-component[latex]\vec A_y[/latex] is the side parallel to the y-axis. Vector component [latex]\vec A_x[/latex] is orthogonal to vector component [latex]\vec A_y[/latex].

It is customary to denote the positive direction on thex-axis by the unit vector i and the positive direction on they-axis by the unit vector j.Unit vectors of the axes, i and j, define two orthogonal directions in the plane. As shown in the figure above, thex– andy– components of a vector can now be written in terms of the unit vectors of the axes:

[latex]\vec A_x = A_x\underline{\hat{i}}[/latex]

[latex]\vec A_y = A_y\underline{\hat{j}}[/latex]

The vectors [latex]\vec A_x and \vec A_y[/latex] defined by the figure aboveare thevector componentsof vector [latex]\vec A[/latex] . The numbers A_xand A_ythat define the vector components above are the scalar components of vector [latex]\vec A[/latex]. Combining the diagram above with the equations above, we obtainthe component form of a vector:

[latex]\vec A=A_x\underline{\hat {i}} + A_y\underline{\hat {j}}[/latex]

If we know the coordinates b(x_b, y_b)of the origin point of a vector (wherebstands for “beginning”) and the coordinates e(x_e, y_e)of the end point of a vector (whereestands for “end”), we can obtain the scalar components of a vector simply by subtracting the origin point coordinates from the end point coordinates:

[latex]A_x = x_e - x_b[/latex]

[latex]A_y = y_e - y_b[/latex]

To specify the location of a point in space, we need three coordinates (x,y,z), where coordinatesxandyspecify locations in a plane, and coordinatezgives a vertical position above or below the plane. Three-dimensional space has three orthogonal directions, so we need not two butthreeunit vectors to define a three-dimensional coordinate system. In the Cartesian coordinate system, the first two unit vectors are the unit vector of thex-axis iand the unit vector of they-axis j. The third unit vector kis the direction of thez-axis, as can be seen below. The order in which the axes are labeled, which is the order in which the three unit vectors appear, is important because it defines the orientation of the coordinate system. The orderx–y–z, which is equivalent to the order i-j-k, defines the standard right-handed coordinate system (positive orientation).

In three-dimensional space, vector [latex]\vec A[/latex] has three vector components: thex-component [latex]\vec A_x = A_x\underline{\hat{i}}[/latex], which is the part of vector[latex]\vec A[/latex] along thex-axis; they-component [latex]\vec A_y = A_y\underline{\hat{j}}[/latex] , which is the part of [latex]\vec A[/latex] along they-axis; and thez-component [latex]\vec A_z = A_z \underline{\hat{k}}[/latex], which is the part of the vector along thez-axis. A vector in three-dimensional space is the vector sum of its three vector components:

[latex]\vec A= A_x\underline{\hat{i}}+A_y\underline{\hat{j}}+A_z\underline{\hat{k}}[/latex]

If we know the coordinates of its origin b(x_b, y_b, z_b)and of its end e(x_e, y_e, z_e) its scalar components are obtained by taking their differences, and thez-component is given by:

[latex]A_z=z_e-z_b[/latex]

MagnitudeAis obtained by the following equation:

[latex]A=\sqrt {A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}[/latex]

This expression for the vector magnitude comes from applying the Pythagorean theorem twice. As seen in the figure below, the diagonal in thex-y plane has length [latex]\sqrt{A_{x}^{2}+A_{y}^{2}}[/latex] and its square adds to the square A_z² to give A² . Note that when thez-component is zero, the vector lies entirely in thex-y plane and its description is reduced to two dimensions.

Source: University Physics Volume 1, OpenStax CNX, https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/2-2-coordinate-systems-and-components-of-a-vector/

Basically: The Cartesian coordinate frame gives us a way to communicate the location of a point in space. In 2-d we use [x, y] and in 3-d we also include z: [x, y, z].

Application: If I am trying to walk across the room, I can walk in a straight line for 5 steps, or I can take 3 steps to the right and 4 steps ahead. If I wanted to describe the position of where I went in the Cartesian coordinate frame, it would be [3, 4] assuming x is to the right and y is straight ahead.

Looking ahead: This will connect with everything we do in this class, especially with cross-products (1.5), torque (1.6), and equilibrium equations (everywhere).