Vectors/Introduction

The basic idea

A vector is a mathematical concept that has both magnitude and direction. Detailed explanation of vectors may be found at the Wikibooks module Linear Algebra/Vectors in Space. In physics, vectors are used to describe things happening in space by giving a series of quantities which relate to the problem's coordinate system.

A vector is often expressed as a series of numbers. For example, in the two-dimensional space of real numbers, the notation (1, 1) represents a vector that is pointed 45 degrees from the x-axis towards the y-axis with a magnitude of ${\displaystyle {\sqrt {2}}}$.

Commonly in physics, we use position vectors to describe where something is in the space we are considering, or how its position is changing at that moment in time. Position vectors are written as summations of scalars multiplied by unit vectors. For example:

${\displaystyle a{\hat {i}}+b{\hat {j}}+c{\hat {k}}}$

where a, b and c are scalars and ${\displaystyle {\hat {i}},{\hat {j}}}$ and ${\displaystyle {\hat {k}}}$ are unit vectors of the Cartesian (René Descartes) coordinate system. A unit vector is a special vector which has magnitude 1 and points along one of the coordinate frame's axes. This is better illustrated by a diagram.

A vector itself is typically indicated by either an arrow: ${\displaystyle {\vec {v}}}$, or just by boldface type: v, so the vector above as a complete equation would be denoted as:

${\displaystyle {\vec {v}}=a{\hat {i}}+b{\hat {j}}+c{\hat {k}}}$

The magnitude of a vector is computed by ${\displaystyle |{\vec {v}}|={\sqrt {\sum _{i}(x_{i}^{2})}}}$. For example, in two-dimensional space, this equation reduces to:

${\displaystyle |{\vec {v}}|={\sqrt {x^{2}+y^{2}}}}$.

For three-dimensional space, this equation becomes:

${\displaystyle |{\vec {v}}|={\sqrt {x^{2}+y^{2}+z^{2}}}}$.

Exercises

Find the magnitude of the following vectors. Answers below.

${\displaystyle {\vec {v}}=(4,3)}$

${\displaystyle |{\vec {v}}|={\sqrt {4^{2}+3^{2}}}=5}$

${\displaystyle {\vec {v}}=(5,3)}$

${\displaystyle |{\vec {v}}|={\sqrt {5^{2}+3^{2}}}={\sqrt {34}}}$

${\displaystyle {\vec {v}}=(1,0)}$

${\displaystyle |{\vec {v}}|={\sqrt {1^{2}+0^{2}}}=1}$

${\displaystyle {\vec {v}}=(4,4)}$

${\displaystyle |{\vec {v}}|={\sqrt {4^{2}+4^{2}}}={\sqrt {32}}}$

${\displaystyle {\vec {v}}=(5,0,0)}$

${\displaystyle |{\vec {v}}|={\sqrt {5^{2}+0^{2}+0^{2}}}=5}$

Using vectors in physics

Many problems, particularly in mechanics, involve the use of two- or three-dimensional space to describe where objects are and what they are doing. Vectors can be used to condense this information into a precise and easily understandable form that is easy to manipulate with mathematics.

Position - or where something is, can be shown using a position vector. Position vectors measure how far something is from the origin of the reference frame and in what direction, and are usually, though not always, given the symbol ${\displaystyle {\vec {r}}}$. It is usually good practice to use ${\displaystyle {\vec {r}}}$ for position vectors when describing your solution to a problem as most physicists use this notation.

Velocity is defined as the rate of change of position with respect to time. You may be used to writing velocity, v, as a scalar because it was assumed in your solution that v referred to speed in the direction of travel. However, if we take the strict definition and apply it to the position vector - which we have already established is the proper way of representing position - we get:

${\displaystyle {\frac {d{\vec {r}}}{dt}}={\frac {da}{dt}}{\hat {i}}+{\frac {db}{dt}}{\hat {j}}+{\frac {dc}{dt}}{\hat {k}}}$

However, we note that the unit vectors are merely notation rather than terms themselves and are in fact not differentated, only the scalars which represent the vector's components in each direction differentiate.

Assuming that each component is not a constant and thus has a non-zero derivative, we get:

${\displaystyle {\vec {v}}=a'{\hat {i}}+b'{\hat {j}}+c'{\hat {k}}}$ where a', b' and c' are simply the first derivatives with respect to time of each original position vector component.

Here it is clear that velocity is also a vector. In the real world this means that each component of the velocity vector indicates how quickly each component of the position vector is changing - that is, how fast the object is moving in each direction.

Vectors in Mechanics

Vector notation is ubiquitous in the modern literature on solid mechanics, fluid mechanics, biomechanics, nonlinear finite elements and a host of other subjects in mechanics. A student has to be familiar with the notation in order to be able to read the literature. In this section we introduce the notation that is used, common operations in vector algebra, and some ideas from vector calculus.

Vectors

A vector is an object that has certain properties. What are these properties? We usually say that these properties are:

• a vector has a magnitude (or length)
• a vector has a direction.

To make the definition of the vector object more precise we may also say that vectors are objects that satisfy the properties of a vector space.

The standard notation for a vector is lower case bold type (for example ${\displaystyle \mathbf {a} \,}$).

In Figure 1(a) you can see a vector ${\displaystyle \mathbf {a} }$ in red. This vector can be represented in component form with respect to the basis (${\displaystyle \mathbf {e} _{1},\mathbf {e} _{2}\,}$) as

${\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}\,}$

where ${\displaystyle \mathbf {e} _{1}\,}$ and ${\displaystyle \mathbf {e} _{2}\,}$ are orthonormal unit vectors. Orthonormal means they are at right angles to each other (orthogonal) and are unit vectors. Recall that unit vectors are vectors of length 1. These vectors are also called basis vectors.

You could also represent the same vector ${\displaystyle \mathbf {a} \,}$ in terms of another set of basis vectors (${\displaystyle \mathbf {g} _{1},\mathbf {g} _{2}\,}$) as shown in Figure 1(b). In that case, the components of the vector are ${\displaystyle (b_{1},b_{2})\,}$ and we can write

${\displaystyle \mathbf {a} =b_{1}\mathbf {g} _{1}+b_{2}\mathbf {g} _{2}\,~.}$

Note that the basis vectors ${\displaystyle \mathbf {g} _{1}\,}$ and ${\displaystyle \mathbf {g} _{2}\,}$ do not necessarily have to be unit vectors. All we need is that they be linearly independent, that is, it should not be possible for us to represent one solely in terms of the others.

In three dimensions, using an orthonormal basis, we can write the vector ${\displaystyle \mathbf {a} \,}$ as

${\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{2}\mathbf {e} _{3}\,}$

where ${\displaystyle \mathbf {e} _{3}\,}$ is perpendicular to both ${\displaystyle \mathbf {e} _{1}\,}$ and ${\displaystyle \mathbf {e} _{2}\,}$. This is the usual basis in which we express arbitrary vectors.