2.9:
Cartesian Vector Notation
Consider a vector A with its x and y components represented in terms of unit vectors, i and j. Here the unit vectors have dimensionless magnitude of one.
Since the magnitude of any vector component is always a positive quantity, represented by scalars, A can be expressed as a Cartesian vector.
Here, a right-handed, rectangle coordinate system is used. The right-hand thumb points toward the positive z-axis, and the fingers curl from the positive x-axis toward the positive y-axis.
A 3-dimensional vector can be represented in rectangular cartesian coordinates using i, j, and k unit vectors. The direction of these vectors are represented depending on the positive or negative axes.
A vector is represented as the vector sum of its individual components, and its magnitude is expressed as the positive square root of the sum of the squares of its components.
Vector algebra operations are simplified by representing vector in the Cartesian form. It separates its magnitude and direction along the axes using unit vector notation.
2.9:
Cartesian Vector Notation
Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This notation represents a vector in terms of three components along the x, y, and z axes, respectively.
For example, suppose we have a vector A pointing in the direction (3, −4, 5). In that case, it can be represented using Cartesian vector notation as A = 3i – 4j + 5k, where i, j, and k are unit vectors along the x, y, and z axes, respectively. The unit vectors are defined as i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1).
Cartesian vector notation can be used to perform various vector operations, such as addition, subtraction, and scalar multiplication. For example, if we have two vectors, A = 3i – 4j + 5k and B = 2i + 7j – 3k, we can add them using Cartesian vector notation as follows:
We can also subtract them as follows:
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp. Page No- 34, 44, 45.