Parallel Vectors & Unit Vectors (Edexcel IGCSE Maths)

Parallel Vectors

Spec point — 8D

How can I tell if two vectors are parallel?

• Two vectors are parallel if one is a scalar multiple of the other

  • This means that all components of the vector have been multiplied by a common constant (scalar)

  • For example, $( \begin{gathered} 1 \\ 3 \end{gathered} )$ and $( \begin{gathered} 2 \\ 6 \end{gathered} )$ are scalar multiples

    • The numbers in the first vector have each been multiplied by 2 to get the numbers in the second vector

• Multiplying the components of a vector by a positive scalar changes the magnitude of the vector but not the direction

  • For example, $( \begin{gathered} 2 \\ 6 \end{gathered} )$ is double the length of $( \begin{gathered} 1 \\ 3 \end{gathered} )$ but in the same direction

• Multiplying the components of a vector by a negative scalar reverses the direction

• You can factorise a vector to help spot if two vectors are parallel

  • $( \begin{gathered} 9 \\ - 3 \end{gathered} ) = 3 ( \begin{gathered} 3 \\ - 1 \end{gathered} )$

  • $( \begin{gathered} - 6 \\ 2 \end{gathered} ) = - 2 ( \begin{gathered} 3 \\ - 1 \end{gathered} )$

    • They are scalar multiples of the same vector so they are parallel

Examples of parallel vectors

Exam Hint If you are told that two vectors ($a$ and $b$) are parallel, then it can be helpful to define a scalar to form an equation $a=kb$ .

Worked Example Show that the vectors $a = ( \begin{gathered} 2 \\ - 4 \end{gathered} )$ and $b = ( \begin{gathered} - 3 \\ 6 \end{gathered} )$are parallel.

Answer:

Method 1

Show that one vector is a multiple of the other

$\begin{gathered} - 1.5 a & = - 1.5 ( \begin{gathered} 2 \\ - 4 \end{gathered} ) \\ = ( \begin{gathered} - 1.5 \times 2 \\ - 1.5 \times - 4 \end{gathered} ) \\ = ( \begin{gathered} - 3 \\ 6 \end{gathered} ) \\ = b \end{gathered}$

Final answer: $b=-1.5a$, so the two vectors are parallel

Method 2

Show that both vectors are multiples of another vector

$a = ( \begin{gathered} 2 \\ - 4 \end{gathered} ) = 2 ( \begin{gathered} 1 \\ - 2 \end{gathered} )$

$b = ( \begin{gathered} - 3 \\ 6 \end{gathered} ) = - 3 ( \begin{gathered} 1 \\ - 2 \end{gathered} )$

Final answer: $a$and$b$are both scalar multiples of$( \begin{gathered} 1 \\ - 2 \end{gathered} )$, so they are parallel to each other

Unit vectors

Spec point — 8D

What is a unit vector?

• A unit vector is a vector with a modulus(length) of 1

• To find a unit vector that is in the same direction as the vector $a$

  • divide the components of the vector by the modulus of the vector
  • I.e. $\frac{a}{\left| a \right|}$ is a unit vector in the direction of vector $a$

• For example, a unit vector in the direction $( \begin{gathered} 3 \\ - 4 \end{gathered} )$ is

$\begin{gathered} \frac{1}{ \sqrt{3^{2} + ( - 4 )^{2}} } ( \begin{gathered} 3 \\ - 4 \end{gathered} ) & = 1 \frac{1}{5} ( \begin{gathered} 3 \\ - 4 \end{gathered} ) \\ = ( \begin{gathered} \frac{3}{5} \\ - \frac{4}{5} \end{gathered} ) \end{gathered}$

Worked Example Find a unit vector in the same direction as $( \begin{gathered} - 2 \\ 5 \end{gathered} )$.

Answer:

Find the modulus of the vector

$\sqrt{( - 2 )^{2} + 5^{2}} = \sqrt{4 + 25} = \sqrt{29}$

Divide each of the vector components by the modulus

$( \begin{gathered} - \frac{2}{ \sqrt{29} } \\ \frac{5}{ \sqrt{29} } \end{gathered} )$