Geometry of Vector Addition & Subtraction (SQA National 5 Maths): Revision Note

Adding & subtracting with directed line segments

How can I represent a vector visually?

• A vector has both a size (magnitude) and a direction

  • You can represent this using a directed line segment

  • You need to draw a line to show the size of the vector

  • You also need to draw an arrowhead to show the direction of the vector

    • The arrowhead can be drawn at the end of the line, or in the middle of the line

• A vector written as $\overrightarrow{\mathrm{AB}}$ will have points $\mathrm{A}$ and $\mathrm{B}$ as its endpoints

  • Here the arrow will point towards B

  • Vector $\overrightarrow{BA}$ will have the same length but will point towards A

How do I draw a vector on a grid?

• You can draw a vector anywhere on a grid

  • Just make sure it has the correct length and the correct direction

• To draw the vector $\mathbf{a}=\left(\begin{array}{c}3\\ 4\end{array}\right)$

  • Pick a point on the grid and draw a dot there

  • Count 3 units to the right and 4 units up and draw another dot

  • Draw a line between the two dots

  • Put an arrow on the line pointing toward the second dot

• Look out for negatives and zeroes

  • $\mathbf{b}=\left(\begin{array}{c}2\\ -4\end{array}\right)$  goes 2 to the right and 4 down

  • $\mathbf{c}=\left(\begin{array}{c}2\\ 0\end{array}\right)$ goes 2 to the right but does not go up or down

What happens when I multiply a vector by a scalar?

• When you multiply a vector by a positive scalar:

  • The direction stays the same

  • The length of the vector is multiplied by the scalar

• For example, $\mathbf{a}=\left(\begin{array}{c}4\\ -2\end{array}\right)$

  • $2\mathbf{a}=\left(\begin{array}{c}8\\ -4\end{array}\right)$ will have the same direction but double the length

  • $\frac{1}{2}\mathbf{a}=\left(\begin{array}{c}2\\ -1\end{array}\right)$ will have the same direction but half the length

• When you multiply a vector by a negative scalar:

  • The direction is reversed

  • The length of the vector is multiplied by the number after the negative sign

• For example, $\mathbf{a}=\left(\begin{array}{c}4\\ -2\end{array}\right)$

  • $-\mathbf{a}=\left(\begin{array}{c}-4\\ 2\end{array}\right)$ will be in the opposite direction and its length will be the same

  • $-2\mathbf{a}=\left(\begin{array}{c}-8\\ 4\end{array}\right)$ will be in the opposite direction and its length will be doubled

How do I represent addition or subtraction of vectors using directed line segments?

• To draw the vector $\mathbf{a}+\mathbf{b}$

  • Draw the vector $\mathbf{a}$

  • Draw the vector $\mathbf{b}$ starting at the endpoint of $\mathbf{a}$

  • Vector $\mathbf{a}+\mathbf{b}$ is the line that starts at the start of $\mathbf{a}$ and ends at the end of $\mathbf{b}$

• To draw the vector $\mathbf{a}-\mathbf{b}$ 

  • Draw the vector $\mathbf{a}$

  • Draw the vector $-\mathbf{b}$  starting at the endpoint of $\mathbf{a}$

  • Vector $\mathbf{a}\mathbf{-}\mathbf{b}$ is the line that starts at the start of $\mathbf{a}$  and ends at the end of $-\mathbf{b}$