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Vectors

Vectors are a mathematical object that has both magnitude and direction.

Conversely, a scalar is a mathematical object that has only magnitude.

This page

This page describes vectors in 3D space, but diagrams are in 2D for simplicity. The thing about vectors that makes them so useful is that they can be used in any number of dimensions, without changing the underlying mathematics.

Vector Notation​

Vectors are typically denoted by a lowercase boldface letter, like .

Alternatively, a vector can be denoted by a letter with an arrow on top, like . Both are valid notations.

Magnitude​

The magnitude of a vector is denoted by , and it is the length of the vector. Sometimes a single bar is used, like , but this can be confused with the absolute value of a scalar.

Addition​

Firstly think of a vector as a displacement in space, like moving by some amount in some direction.

Adding two vectors is like moving by the first vector and then moving by the second vector.

Addition is commutative, meaning . This can be visually seen:

The red and blue vectors are the original vectors, and the green vector is the sum of the two. You can see that the green vector is the same no matter which order you add the red and blue vectors.

The addition of vectors is also associative, meaning .

Identity Element​

The identity element for vector addition is the zero vector, denoted by . The zero vector has a magnitude of 0 and is the only vector that has a magnitude of 0.

Formally, . This means that adding the zero vector to any vector gives the original vector.

Inverse Element​

There's also an inverse element for vector addition, denoted by . The inverse element is the vector that, when added to the original vector, gives the zero vector.

Formally, .

Conceptually, it's like going in some direction and then going back in the opposite direction.

Scalar Multiplication​

Scalar multiplication is multiplying a vector by a scalar (a number). Multiplying by the magnitude of a vector is the same as the magnitude of a vector multiplied by a scalar. Meaning, .

Scalar multiplication is associative, meaning .

Distributive Law​

Scalar multiplication is distributive over vector addition, meaning .

Identity Element for Scalar Multiplication​

The identity element for scalar multiplication is 1. This means that .

Coordinate Systems​

Coordinate systems are used to describe the position of vectors in space. A coordinate system can be broken down into these components:

  1. Origin: The point where all axes intersect.
  2. Axes: Lines that extend from the origin in different directions. For example , , and axes.
  3. Positive Direction: The direction in which the axes increase (more positive).
  4. Basis Vectors: Vectors of length 1 that point in the positive direction of each axis.

Cartesian Coordinates​

Cartesian coordinates are the most common coordinate system. The components of the Cartesian coordinate system include:

  1. Origin: denoted as .
  2. Axes: , , and axes.
  3. Positive Direction: We are free to choose the positive direction of each axis. In physics it's best to choose it such that it best fits a given problem. Many problems that are difficult to solve in the conventional choices can be solved much more easily by choosing different axes.

For the basis vectors, let be a point in space. For that point, the basis vectors are:

  • : The basis vector in the direction of the -axis.
  • : The basis vector in the direction of the -axis.
  • : The basis vector in the direction of the -axis.

If another point is chosen, the basis vectors remain the same; , , and . Therefore, the references are dropped and the basis vectors are denoted as , , and .

Cylindrical Coordinates​

Cylindrical coordinates are a different way to describe the position of a point in space. The components of the cylindrical coordinate system include:

  1. Origin: just like in Cartesian coordinates, denoted as .
  2. Axes: , , and axes. The axis is the distance from the origin, is the angle from the -axis, and is the same as the axis in Cartesian coordinates.
  3. Positive Direction: The positive direction of the axis is outwards from the origin, the positive direction of the axis is counterclockwise from the -axis, and the positive direction of the axis is the same as the axis in Cartesian coordinates.

Unlike Cartesian coordinates, the basis vectors in cylindrical coordinates change based on the point. Hence, references are kept, and for point , the basis vectors are:

  • : The basis vector in the direction of the -axis.
  • : The basis vector in the direction of the -axis.
  • : The basis vector in the direction of the -axis.

(Diagram coming soon)

In this system, the and components are polar coordinates. To convert between Cartesian and cylindrical coordinates, trigonometry is used:

Likewise,

Vector Product (Cross Product)​

The cross product of two vectors and is denoted by .