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: