Column Vectors: Writing Vectors as East–North Components

1. What is a column vector?

On a graph, East is to the right and North is up. A column vector has two numbers in a bracket, one above the other:

$\left(\begin{array}{c}x\\ y\end{array}\right)$ where x is the movement along the East–West line and y is the movement along the North–South line.

• Top number x: positive means East, negative means West.
• Bottom number y: positive means North, negative means South.

For example, $\left(\begin{array}{c}3\\ 4\end{array}\right)$ means 3 units East and 4 units North. The vector $\left(\begin{array}{c}-4\\ 3\end{array}\right)$ means 4 units West and 3 units North. To draw it on squared paper, start at any point, go across by the top number, then up or down by the bottom number, and join the start to the finish with an arrow.

2. From a bearing to a column vector using a graph

Take the vector (5 km, 030°). We can find its components by drawing it accurately on squared paper with a scale, say 1 cm to 1 km, and measuring.

1. 1Mark the starting point A. Draw the North line upwards.
2. 2Measure 030° clockwise from North with a protractor and draw a line 5 cm long to B.
3. 3From B, draw a line straight down to the East line through A. This makes a right-angled triangle.
4. 4Measure the horizontal side: 2.5 cm, so 2.5 km East. Measure the vertical side: 4.3 cm, so about 4.3 km North.
5. 5Write the column vector: $\left(\begin{array}{c}2.5\\ 4.3\end{array}\right)$

▶ See how the vector (5 km, 030°) splits into an East part and a North part.

Do the same for (25 km, 150°). The angle is between 090° and 180°, so the vector points South-East: the East part is positive and the North–South part is negative (South). With a scale of 1 cm to 5 km, the line is 5 cm long. Measurement gives about 2.5 cm East and 4.3 cm South, so the vector is approximately $\left(\begin{array}{c}12.5\\ -21.7\end{array}\right)$.

▶ A vector pointing South-East: East is positive, South is negative.

3. Finding the components without a graph

Every vector (d, θ) makes a right-angled triangle with the North line. Look at the triangle in the animation: the side along the East–West line is opposite the bearing angle, and the side along the North–South line is adjacent to it. With the bearing measured from North:

East component x = d × sin θ
North component y = d × cos θ

Check with (5 km, 030°): x = 5 × sin 30° = 5 × 0.5 = 2.5 and y = 5 × cos 30° = 5 × 0.866 = 4.33. This agrees with our drawing. Calculators automatically give the correct signs when you use the full bearing: for 150°, x = 25 sin 150° = 12.5 (positive, East) and y = 25 cos 150° = −21.65 (negative, South).

Another example: (8 km, 200°). x = 8 sin 200° = −2.74 and y = 8 cos 200° = −7.52. So the vector is $\left(\begin{array}{c}-2.74\\ -7.52\end{array}\right)$: 2.74 km West and 7.52 km South, which agrees with 200° being in the South-West.

▶ Both components are negative because 200° points into the South-West.

4. Vectors along the axes

Some bearings need no calculation at all because the vector lies along an axis. Due East (090°) has no North–South part; due North (000°) has no East–West part.

For (10 km, 270°) we get x = 10 sin 270° = −10 and y = 10 cos 270° = 0. For (70 km, 090°): x = 70 sin 90° = 70 and y = 70 cos 90° = 0. These match the picture: 10 km West and 70 km East.

The magnitude of a vector given as (d, θ) is just d, and its direction is θ. For column vectors along an axis, the magnitude is the size of the nonzero number: $\left(\begin{array}{c}-10\\ 0\end{array}\right)$ has magnitude 10 km and direction 270° (West).

Worked examples

Try each one yourself first, then open the solution and compare every step.