4. Vectors

Abstract

Vectors provide a powerful tool for describing quantities that have magnitude and direction. A simple example of such a quantity is force: when applying a force, one is generally interested in how much force is applied (i.e., the magnitude of the force) and the direction in which the force was applied. Analysis of ship’s stability arises from the effect of upsetting forces or heeling moments, which can be represented as vectors.

This chapter begins with the definition of the concept of vector, and then the basic operations including vectors are defined. Scalar, vector and mixed product as well as their applications will be presented. Exercises are also included, with the answers at the end of each task.

Aim

To acquire skills in solving tasks with vectors, and also to understand concepts standing behind those calculations. Vectors are also important for a number of real-life applications in the maritime field, which can be seen from the last part.

Learning outcomes

View vectors geometrically
Determine equal and opposite vectors, magnitude and unit vector
Express vector as linear combination of vectors
Demonstrate vectors in the coordinate system
Calculate with vectors algebraically and graphically
Apply scalar, vector and mixed vector product.

Previous knowledge of mathematics

determinants, geometry in the plane R² and space R³

Relatedness with solving problems in the maritime field

The real-world problems encountered most often with vectors are navigation problems. These navigation problems use variables like speed or velocity of vessel and direction or course to form vectors for computation. Some navigation problems ask us to find the actual (ground) speed of a vessel in wind situation using the combined forces of the wind and the vessel’s velocity. Additionally, problems such as plotting, effect of ocean current on navigation, Cremona diagram used in statics of trusses to determine the forces in members can be solved by vector algebra.

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