Introduction to Column Vectors (Edexcel IGCSE Maths B): Revision Note

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

  • $\left(\begin{array}{c}6\\ 3\end{array}\right)$ 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

    • $\left(\begin{array}{c}5\\ 2\end{array}\right)+\left(\begin{array}{c}3\\ -1\end{array}\right)=\left(\begin{array}{c}5+3\\ 2+\left(-1\right)\end{array}\right)=\left(\begin{array}{c}8\\ 1\end{array}\right)$

• To subtract column vectors

  • Subtract the second top number from the first

  • Subtract the second bottom number from the first

    • $\left(\begin{array}{c}5\\ 2\end{array}\right)-\left(\begin{array}{c}3\\ -1\end{array}\right)=\left(\begin{array}{c}5-3\\ 2-\left(-1\right)\end{array}\right)=\left(\begin{array}{c}2\\ 3\end{array}\right)$

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

    • $3\left(\begin{array}{c}2\\ -1\end{array}\right)=\left(\begin{array}{c}3\times 2\\ 3\times \left(-1\right)\end{array}\right)=\left(\begin{array}{c}6\\ -3\end{array}\right)$

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

• You need to follow the order of operations

  • $2\left(\begin{array}{c}5\\ 2\end{array}\right)+5\left(\begin{array}{c}3\\ -1\end{array}\right)$

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

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

• STEP 2
  Add or subtract the new column vectors

  • $\left(\begin{array}{c}10+15\\ 4+\left(-5\right)\end{array}\right)=\left(\begin{array}{c}25\\ -1\end{array}\right)$

How do I multiply a vector by a matrix?

• Note that a column vector may also be thought of as a $\mathbf{2}\mathbf{\times }\mathbf{1}$ matrix

  • I.e. a matrix with 2 rows and 1 column

• As such, it may be multiplied by a $2\times 2$ matrix

  • $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}ax+by\\ cx+dy\end{array}\right)$

• See the revision notes on multiplying matrices and transformations with matrices for more details on this