Vectors

Under construction This section is incomplete and may contain errors.

A vector is a group of components, each of which is a scalar (number). 2D and 3D maths make use of these to represent positions, velocities and related concepts.

Each component of a vector represents a ‘distance’ along some axis, for example in the 2D case, the vector $(1 2)$ means:

• 1 unit along the first component (x)
• 2 units along the second component (y)

The physical quanity is not important (ie, it does not matter if ‘unit’ means meters, kilometers, pixels, inches, height of Joe Daweses), but they should be the same.

The components can be thought of as a ‘movement’ away from one point to another. If the starting point is $(0,0)$, the vector can be seen ‘distance travelled’ from the origin of a graph:

Vector Notation

There are many conventions for how to represent vectors, lower ase letters with an arrow above them (eg, $\vec{a}$), or bold (eg, $\mathbf{a}$).

The components of the vectors are sometimes represented as subscripts (small letters) on the vector, for a vector named $a$:

• the first component might be represented as $a_x$
• the second component as $a_y$.

The components may also be referred by their numeric index, $a_x$ might instead be represented as as $a_1$, and $a_y$ might be represented as $a_2$. This is especially common when using summation or matries.

A vector between two points (A and B) is sometimes represented as the two points with an arrow above them (eg, $\vec{AB}$).

Unit vector

A vector with a magnitude of 1 is referred to a unit vector. A non-unit vector can be coverted into a unit vector by normalising it (see vector operations below).

Vector Operations

There are a number of vector operations which are useful to know.

Addition and Subtraction

Vectors can be added together, doing so results in the corrisponding components of the vectors being added together:

$$ \vec{c} = \vec{a} + \vec{b} $$

Is calculated as:

$$\begin{aligned} \textbf{c}_1 &= \textbf{a}_1 + \textbf{b}_1 \\ \textbf{c}_n &= \textbf{a}_n + \textbf{b}_n \end{aligned} $$

Magnitude

The magnitude of a vector is found by square rooting the sum of each of its components squared. The magnitude of a vector is represented by the vector in a pair of vertical bars ($ \lVert\mathbf{a}\rVert$).

2D:

$ \lVert\mathbf{a}\rVert = \sqrt{{a_x}^2 + {a_y}^2} $

3D:

$ \lVert\mathbf{b}\rVert = \sqrt{{a_x}^2 + {a_y}^2 + {a_z}^2} $

Normalised Vectors

A vector of length 1, can be found by dividing a vector by its length

See Also

• BBC Bitesize - Vectors