Vector Maths

Vector equality

Two vectors are said to be equal if they have the same magnitude and direction. However if we are talking about free vectors, then two free vectors are equal if they have the same base point and end point.  For example, the vector e₁ + 2e₂ + 3e₃ with base point (1,0,0) and the vector e₁+2e₂+3e₃ with base point (0,1,0) are different free vectors, but the same (displacement) vector.

Vector addition and subtraction

Let a=a₁e₁ + a₂e₂ + a₃e₃ and b=b₁e₁ + b₂e₂ + b₃e₃, where e₁, e₂, e₃ are orthogonal unit vectors (Note: they only need to be linearly independent, i.e. not parallel and not in the same plane, for these algebraic addition and subtraction rules to apply)

The sum of a and b is:

The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below:

This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are free vectors, then the addition is only defined if a and b have the same base point, which will then also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c).

The difference of a and b is:

Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below:

If a and b are free vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference. This operation deserves the name "subtraction" because (a − b) + b = a.

Scalar multiplication

A vector may also be multiplied, or re-scaled, by a real number r. In the context of spatial vectors, these real numbers are often called scalars (from scale) to distinguish them from vectors. The operation of multiplying a vector by a scalar is called scalar multiplication. The resulting vector is:

Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. Geometrically, this can be visualized (at least in the case when r is an integer) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.

If r is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (r = -1 and r = 2) are given below:

Scalar multiplication is distributive over vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b.

The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a vector space. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated.

In physics, scalars may also have a unit of measurement associated with them. For instance, Newton's second law is

where F has units of force, a has units of acceleration, and the scalar m has units of mass. In one possible physical interpretation of the above diagram, the scale of acceleration is, for instance, 2 m/s² : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.

Length of a vector

The length or magnitude or norm of the vector a is denoted by ||a|| or, less commonly, |a|, which is not to be confused with the absolute value (a scalar "norm").

The length of the vector a = a₁e₁ + a₂e₂+ a₃e₃ in a three-dimensional Euclidean space, where e₁, e₂, e₃ are orthogonal unit vectors, can be computed with the Euclidean norm

which is a consequence of the Pythagorean theorem since the basis vectors e₁ , e₂ , e₃ are orthogonal unit vectors.

This happens to be equal to the square root of the dot product of the vector with itself:

Vector length and units

If a vector is itself spatial, the length of the arrow depends on a dimensionless scale. If it represents e.g. a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1:250 and 1 m:50 N respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance.

Unit vector

A unit vector is any vector with a length of one; geometrically, it indicates a direction but no magnitude. If you have a vector of arbitrary length, you can divide it by its length to create a unit vector. This is known as normalizing a vector. A unit vector is often indicated with a hat as in â.

To normalize a vector a = [a₁, a₂, a₃], scale the vector by the reciprocal of its length ||a||. That is:

Dot product

The dot product of two vectors a and b (sometimes called the inner product, or, since its result is a scalar, the scalar product) is denoted by a ∙ b and is defined as:

where ||a|| and ||b|| denote the norm (or length) of a and b, and θ is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a.

The dot product can also be defined as the sum of the products of the components of each vector:

where a and b are vectors of n dimensions; a₁, a₂, …, a_n are coordinates of a; and b₁, b₂, …, b_n are coordinates of b.

This operation is often useful in physics; for instance, work is the dot product of force and displacement.

Cross product

The cross product (also called the vector product or outer product) differs from the dot product primarily in that the result of the cross product of two vectors is a vector. While everything that was said above can be generalized in a straightforward manner to more than three dimensions, the cross product is only meaningful in three dimensions, although the seven dimensional cross product is similar in some respects. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as:

where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b. The problem with this definition is that there are two unit vectors perpendicular to both b and a.

The cross product a × b is defined so that a, b, and a × b also becomes a right handed system (but note that a and b are not necessarily orthogonal). This is the right-hand rule.

The length of a × b can be interpreted as the area of the parallelogram having a and b as sides.

The cross product of two vectors is a pseudovector (see below).

Scalar triple product

The scalar triple product (also called the box product or mixed triple product) is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by (a b c) and defined as:

It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are right-handed.

In components ( with respect to a right-handed orthonormal basis), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense: