# General Vectors

# 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.