Scalar and vector quantities

Throughout these pages we are going to use two types of quantities: scalar and vector.

A scalar quantity is fully described by a magnitude or numerical value and its units.

A vector quantity is described by a magnitude (with its units) and its direction.

A physical quantity will be defined as a scalar or as a vector depending on its nature. When a quantity is specified with its magnitude (or size) alone, and its units, we will use a scalar. For example temperature, density, volume, mass… On the other hand, when a quantity needs to have a direction to be fully specified, we will define it as a vector. For example force, position vector, velocity, acceleration, electric field… these are all quantities which have a direction.

A scalar quantity is represented by a single letter and a vector quantity is represented by a letter with an arrow over it (sometimes by a bold letter without the arrow).

Vector quantities. Cartesian coordinates

A vector quantity or vector is a directed line segment.

In order to work with vectors we have to be able to quantify their magnitude (or modulus) as well as their orientation with respect to a coordinate system.

In these pages we will mostly use the Cartesian coordinate system. In three dimensions it consists of three perpendicular oriented lines (or x, y, z axes). The point where they meet is called the origin. In two dimensions the Cartesian coordinate system has two axes. As we will see, the positive direction of each axis is usually represented by a unit vector .

In the previous figure you can see a vector a in two dimensions. The Cartesian components of this vector, which we will call a_x and a_y, are the orthogonal projections of this vector onto the x– and y-axis, respectively.

The Cartesian components of the vector sketched in the figure above are 5 and 4 respectively, so this vector a is given by:

From the Cartesian components of a vector we can derive its magnitude (sometimes called modulus) and the angle from the x-axis. These two numbers are the polar coordinates of the vector.

From the above figure and using the Pithagoras’ theorem:

The magnitude of a vector is shown by two vertical bars on either side of the vector or by the letter representing the vector without the arrow. Both forms are equivalent.

Now, using the trigonometric functions sine, cosine and tangent:

Using the equations above we can convert either from Cartesian to polar coordinates or the other way round.

For the vector a sketched in the figure above:

You can see that, due to the triangular inequality, the component vectors don’t add up to the magnitude of the main vector:

It is important to use proper notation. Don’t forget to express vectors with an arrow over them.

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