General Vectors

What is a Vector?

A vector is a quantity that has both magnitude (a length) and direction. It is usually represented by a line segment directed from one point to another.
Unlike scalars, which only have magnitude, vectors account for both how much and where.
Vectors can exist in any number of dimensions; however, one, two, or three dimensions are typical in A Level mathematics.

Vector Notation

Vectors are commonly denoted by lowercase boldface letters such as a, b, or v.
The notation used to denote the vector from point A to point B is often AB.
Alternatively, vectors can be represented with i, j, k components. For example, a vector in three dimensions can be depicted as ai + bj + ck.

Vector Arithmetics

Addition of Vectors: To add two vectors, you ‘place them head to tail’ and draw the resultant vector from the tail of the first vector to the head of the second.
Subtraction of Vectors: To subtract a vector, you simply add its negative or ‘opposite’. This involves reversing the direction of the vector you are subtracting and then adding.
Multiplication by Scalars: When a vector is multiplied by a scalar, the magnitude of the vector changes but the direction remains the same (unless the scalar is negative, which reverses the direction).

Unit Vectors

A unit vector is a vector of length 1. It typically represents the direction of a vector.
i, j and k are the standard unit vectors in the x, y and z directions respectively.
Any vector can be converted to a unit vector by dividing the vector by its own magnitude.

Vector Applications

Vector concepts are used in a variety of fields, including physics, engineering, computer graphics and navigation.
In physics, for example, vectors are useful for representing velocity, force and displacement.
In areas like computer graphics, vectors are necessary to render three-dimensional images.