Revision of N5 Vectors

Magnitude

A vector is a quantity with both magnitude and direction. As a result, we can describe vectors as having 'directional growth'.

A vector can be named in one of the following ways:

• Using the letters at the ends of a directed line segment: $\vec{AB}$
• Using a lowercase letter underlined (sometimes in bold if typed): $\vec{u}$ or u

Components

A vector can be written in component form:

e.g. $\begin{pmatrix} 2 \\ 7 \end{pmatrix}$ means move 2 units in the x-direction and 7 units in the y-direction.

e.g. $\begin{pmatrix} 2 \\ 7 \\ 3 \end{pmatrix}$ means move 2 units in the x-direction, 7 units in the y-direction and 3 units in the z-direction.

Magnitude

The magnitude or length of a 2-dimensional vector is found in a similar way we would find the distance between two points.

Let $\vec{u} = \begin{pmatrix} a \\ b \end{pmatrix}$, then $|\vec{u}| = \sqrt{a^2+b^2}$

In 3-dimensions:

Let $\vec{u} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$, then $|\vec{u}| = \sqrt{a^2+b^2+c^2}$

Equal Vectors

Vectors are equal if they have the same magnitude and direction (directional growth) i.e. all of their components are equal.

Conversely, if all components of vectors are equal then the vectors have the same magnitude and direction.

If $\begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} p \\ q \\ r \end{pmatrix}$ then $a=p$, $b=q$ and $c=r$.

Additive Properties of Vectors

Since vectors have directional growth, adding vectors means we are accumulating the growth contained in several vectors.

If $\vec{u} = \begin{pmatrix} 8 \\ -2 \\ 0 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} 4 \\ 6 \\ -3 \end{pmatrix}$, then $\vec{u} + \vec{v} = \begin{pmatrix} 8 \\ -2 \\ 0 \end{pmatrix} + \begin{pmatrix} 4 \\ 6 \\ -3 \end{pmatrix} = \begin{pmatrix} 12 \\ 4 \\ -3 \end{pmatrix}$.

If $\vec{u} = \begin{pmatrix} 8 \\ -2 \\ 0 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} 4 \\ 6 \\ -3 \end{pmatrix}$, then $\vec{u} - \vec{v} = \begin{pmatrix} 8 \\ -2 \\ 0 \end{pmatrix} - \begin{pmatrix} 4 \\ 6 \\ -3 \end{pmatrix} = \begin{pmatrix} 4 \\ -8 \\ 3 \end{pmatrix}$.

Multiplying by a Scalar

Since vectors have directional growth, multiplying an existing vector by a scalar means we are making this existing vector stronger (in the same direction). If you multiply by a negative, then the direction of the vector will change.

If $\vec{u} = \begin{pmatrix} 8 \\ -2 \\ 0 \end{pmatrix}$, then $3\vec{u} = 3\begin{pmatrix} 8 \\ -2 \\ 0 \end{pmatrix} = \begin{pmatrix} 24 \\ -6 \\ 0 \end{pmatrix}$.