Vector Notation

Many quantities in technology cannot be described fully without giving their direction as well as their magnitude (how large a quantity is relative to zero). For example, it is not always useful to know how fast something is moving (speed) without knowing the direction in which it is moving in.

Velocity is called a vector quantity because velocity gives you both magnitude and direction, as opposed to speed, which gives you only the magnitude – called a scalar quantity. Quantities that have magnitude but no direction are called scalar quantities. This includes time, volume, mass, and so on.

Apart from velocity, other vector quantities include force and acceleration. A vector is represented by an arrow whose length is proportional to the magnitude of the vector and whose direction is the same as the direction (i.e. in degrees or radians) of the vector quantity.

This unit will introduce how to add individual vectors to produce a single vector, known as a resultant vector. Resultant vectors can also be broken down into component vectors using trigonometric functions and the Pythagorean theorem. Resolving a vector means to replace it by its components (more on this next section).

Before you can learn that, you need to recognize how vectors are communicated on paper. Vectors are represented differently in different textbooks, but they are usually written in boldface type. The most common notation used in textbooks have the following characteristics:

  • Boldface roman capitals,
  • and non-boldface italic capitals to represent scalar quantities.

For this lesson, B is understood to be a vector quantity, having both magnitude and direction, while B is understood to be a scalar quantity, having magnitude but no direction. Taking into account the vector arrow shown earlier, it can also be represented as:


When written by hand, a vector accent (⇀) is added above the letters:


Similarly, scalar quantities written by hand are surrounded by the absolute symbol (| |) or sometimes with double absolute values (|| ||) with a vector accent:


Representing Vectors in Print and on Paper

You can specify a vector’s magnitude and direction by writing it in polar form. For example, to write a vector that has a length of 5 units and a direction of 38°:


On a Cartesian plane, this would look like: