Introduction to Column Vectors (Cambridge International IGCSE Mathematics)

Basic Vectors

What are column vectors?

• A column vector can be used to describe how to get from one point to another point

  • This is also called a translation vector
  • $( \begin{gathered} 6 \\ 3 \end{gathered} )$ means 6 units to the right and 3 units up

How do I add and subtract column vectors?

• Adding and subtracting vectors is done by looking at the top numbers and bottom numbers separately

• To add column vectors

  • Add the top numbers together

  • Add the bottom numbers together

    • $( \begin{gathered} 5 \\ 2 \end{gathered} ) + ( \begin{gathered} 3 \\ - 1 \end{gathered} ) = ( \begin{gathered} 5 + 3 \\ 2 + ( - 1 ) \end{gathered} ) = ( \begin{gathered} 8 \\ 1 \end{gathered} )$

• To subtract column vectors

  • Subtract the second top number from the first

  • Subtract the second bottom number from the first

    • $( \begin{gathered} 5 \\ 2 \end{gathered} ) - ( \begin{gathered} 3 \\ - 1 \end{gathered} ) = ( \begin{gathered} 5 - 3 \\ 2 - ( - 1 ) \end{gathered} ) = ( \begin{gathered} 2 \\ 3 \end{gathered} )$

How do I multiply a vector by a scalar?

• A scalar is a number not a vector

  • It does not have a direction

• To multiply a column vector by a scalar

  • Multiply the top number by the scalar

  • Multiply the bottom number by the scalar

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

How do I write an expression as a single column vector?

• You need to follow the order of operations

  • $2 ( \begin{gathered} 5 \\ 2 \end{gathered} ) + 5 ( \begin{gathered} 3 \\ - 1 \end{gathered} )$

• STEP 1 Multiply each vector by the scalar in front of it

  • $( \begin{gathered} 2 \times 5 \\ 2 \times 2 \end{gathered} ) + ( \begin{gathered} 5 \times 3 \\ 5 \times ( - 1 ) \end{gathered} ) = ( \begin{gathered} 10 \\ 4 \end{gathered} ) + ( \begin{gathered} 15 \\ - 5 \end{gathered} )$

• STEP 2 Add or subtract the new column vectors

  • $( \begin{gathered} 10 + 15 \\ 4 + ( - 5 ) \end{gathered} ) = ( \begin{gathered} 25 \\ - 1 \end{gathered} )$

Worked Example $a = ( \begin{gathered} p \\ 3 \end{gathered} )$ and $b = ( \begin{gathered} - 2 \\ 1 \end{gathered} )$.

Given that $2 a + 3 b = ( \begin{gathered} 4 \\ q \end{gathered} )$, find the value of $p$ and the value of $q$.

Answer:

Write the left-side side as one vector Multiply each vector by the scalar in front of it

$( \begin{gathered} 2 p \\ 6 \end{gathered} ) + ( \begin{gathered} - 6 \\ 3 \end{gathered} ) = ( \begin{gathered} 4 \\ q \end{gathered} )$

Add the vectors together

$( \begin{gathered} 2 p - 6 \\ 9 \end{gathered} ) = ( \begin{gathered} 4 \\ q \end{gathered} )$

The top components are equal Form and solve an equation

$\begin{gathered} 2 p - 6 & = 4 \\ 2 p & = 10 \\ p = 5 \end{gathered}$

The bottom components are equal

$9=q$

$p=5$ and $q=9$