Representing Vectors as Diagrams (Cambridge International IGCSE Mathematics)

Vector Diagrams

How can I represent a vector visually?

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

  • You need to draw a line to show the size of the vector
  • You also need to draw an arrow to show the direction of the vector

• Vectors are written in bold when typed to show that they are a vector and not a scalar

  • When writing a vector in an exam you should underline the letter to show it is a vector

  • $a$ when typed and $\underline{a}$ when handwritten

    • You will not lose marks if you forget to underline vectors

• If a vector starts at A and ends at B we can write it as $\overset{\rightarrow}{AB}$

  • Here the arrow will point toward B
  • Vector $\overset{\rightarrow}{BA}$ will have the same length but point toward 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 $a = ( \begin{gathered} 3 \\ 4 \end{gathered} )$

  • 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

  • $b = ( \begin{gathered} 2 \\ - 4 \end{gathered} )$ goes 2 to the right and 4 down
  • $c = ( \begin{gathered} 2 \\ 0 \end{gathered} )$ 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, $a = ( \begin{gathered} 4 \\ - 2 \end{gathered} )$

  • $2 a = ( \begin{gathered} 8 \\ - 4 \end{gathered} )$ will have the same direction but double the length
  • $1 \frac{1}{2} a = ( \begin{gathered} 2 \\ - 1 \end{gathered} )$ 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, $a = ( \begin{gathered} 4 \\ - 2 \end{gathered} )$

  • $- a = ( \begin{gathered} - 4 \\ 2 \end{gathered} )$ will be in the opposite direction and its length will be the same
  • $- 2 a = ( \begin{gathered} - 8 \\ 4 \end{gathered} )$ will be in the opposite direction and its length will be doubled

What happens when I add or subtract vectors?

• To draw the vector $a+b$

  • Draw the vector $a$
  • Draw the vector $b$ starting at the endpoint of $a$
  • Draw a line that starts at the start of $a$ and ends at the end of $b$

• To draw the vector $a-b$

  • Draw the vector $a$
  • Draw the vector $-b$ starting at the endpoint of $a$
  • Draw a line that starts at the start of $a$ and ends at the end of $-b$

Worked Example The points A, B and C have coordinates (-4, 2), (2, 4) and (3, -4), respectively.

(a)

Write the vectors $\overset{\rightarrow}{AB},\overset{\rightarrow}{AC}$ and $\overset{\rightarrow}{CB}$ as column vectors.

Answer:

Start by drawing the three vectors onto a grid

From A to B, it is 6 to the right and 2 up

A B -vec = stretchy ( | 6 | 2 | stretchy )

From A to C, it is 7 to the right and 6 down

A C -vec = stretchy ( | 7 | - 6 | | stretchy )

From C to B, it is 1 to the left and 8 up

C B -vec = stretchy ( | - 1 | | 8 | stretchy )

(b)

Without using any calculations, explain why $\vec{AB} + \vec{BC} + \vec{CA} = ( \begin{gathered} 0 \\ 0 \end{gathered} )$.

Answer:

The vector goes from A to B, then from B to C, then from C back to A

Final answer: The vector returns to its starting point