# 14.1: Introduction to Vectors

In this chapter we will review a few concepts you probably know from your physics courses. This chapter does not intend to cover the topic in a comprehensive manner, but instead touch on a few concepts that you will use in your physical chemistry classes.

A vector is a quantity that has both a magnitude and a direction, and as such they are used to specify the position, velocity and momentum of a particle, or to specify a force. Vectors are usually denoted by boldface symbols (e.g. $$\mathbf{u}$$) or with an arrow above the symbol (e.g. $$\vec{u}$$). A tilde placed above or below the name of the vector is also commonly used in shorthand ($$\widetilde{u}$$,$$\underset{\sim}{u}$$).

If we multiply a number $$a$$ by a vector $$\mathbf{v}$$, we obtain a new vector that is parallel to the original but with a length that is $$a$$ times the length of $$\mathbf{v}$$. If $$a$$ is negative $$a\mathbf{v}$$ points in the opposite direction than $$\mathbf{v}$$ . We can express any vector in terms of the so-called unit vectors. These vectors, which are designated $$\hat{\mathbf{i}}$$, $$\hat{\mathbf{j}}$$ and $$\hat{\mathbf{k}}$$, have unit length and point along the positive $$x, y$$ and $$z$$ axis of the cartesian coordinate system (Figure $$\PageIndex{1}$$). The symbol $$\hat{\mathbf{i}}$$ is read "i-hat". Hats are used to denote that a vector has unit length.

The length of $$\mathbf{u}$$ is its magnitude (or modulus), and is usually denoted by $$u$$:

$\label{eq:vectors1} u=|u|=(u_x^2+u_y^2+u_z^2)^{1/2}$

If we have two vectors $$\mathbf{u}=u_x\hat{\mathbf{i}}+u_y \hat{\mathbf{j}}+u_z \hat{\mathbf{k}}$$ and $$\mathbf{v}=v_x \hat{\mathbf{i}}+v_y \hat{\mathbf{j}}+v_z \hat{\mathbf{k}}$$, we can add them to obtain

$\mathbf{u}+\mathbf{v}=(u_x+v_x)\hat{\mathbf{i}}+(u_y+v_y)\hat{\mathbf{j}}+(u_z+v_z)\hat{\mathbf{k}} \nonumber$

or subtract them to obtain:

$\mathbf{u}-\mathbf{v}=(u_x-v_x)\hat{\mathbf{i}}+(u_y-v_y)\hat{\mathbf{j}}+(u_z-v_z)\hat{\mathbf{k}} \nonumber$

When it comes to multiplication, we can perform the product of two vectors in two different ways. The first, which gives a scalar (a number) as the result, is called scalar product or dot product. The second, which gives a vector as a result, is called the vector (or cross) product. Both are important operations in physical chemistry.