Year 11 Extended · Unit 7 Networks and vectors · Lesson 4 of 8

How do you describe a movement with two numbers?

A column vector records how far a movement goes across and how far it goes up. This lesson finds how long that movement is and which way it points, and shows that the same vector can be drawn starting anywhere.

Geometry and TrigonometryScientific calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: a right-angled triangle has shorter sides 6 and 8. How long is the hypotenuse?
    Answer10. 36 + 64 = 100.
  2. From Year 10: what is tan−1(1)?
    Answer45 degrees.
  3. From lesson 3: a tree has 9 vertices. How many edges?
    Answer8.
Words used in this lesson
vector
a quantity with a size and a direction. For example: a movement of 3 across and 4 up.
column vector
a vector written as two numbers one above the other. For example: across over up.
magnitude
the length of a vector. For example: how long the arrow is.
direction
the angle a vector makes with the positive x-axis. For example: measured anticlockwise.
displacement
the vector from a start point to an end point. For example: from home to school.

How far has the drone moved, and in which direction?

A drone in a paddy field near Hue is flown 60 metres east and then 80 metres north. A single straight flight would have given the same result. To describe that single movement you need two things: how big it is and which way it points. A vector carries both. Written as a column vector it lists the movement across on top and the movement up beneath, such as 60
80.

Predict first: will the straight-line distance be more or less than 60 + 80 = 140 metres, and by how much? Then use the figure to see the magnitude and the direction of a column, and check whether two vectors of the same size must point the same way.

5
−2
5.4magnitude
338.2direction in degrees from the positive x-axis

The vector (5, −2) goes 5 to the right and 2 down, so its magnitude is 5.4 and its direction is 338.2 degrees.

Show me the steps
  1. The column (3, 4) goes 3 right and 4 up: magnitude is the square root of 9 + 16, which is 5, and the direction is 53.1 degrees.
  2. The column (−3, 4) has the same magnitude 5, but it goes left and up, so the direction is 180 − 53.1 = 126.9 degrees.
  3. The column (3, −4) has magnitude 5 and goes right and down, so the direction is 360 − 53.1 = 306.9 degrees.
  4. Any column (x, y): the magnitude is the square root of x squared plus y squared, and the direction is the inverse tangent of y over x, adjusted for the quadrant.

A column vector x
y says how far across and how far up. It has no fixed position: the same column can be drawn starting at any point. The vector from A to B is the coordinates of B minus the coordinates of A.

Magnitude is the length: |a| = √(x2 + y2), by Pythagoras. Direction is the angle measured anticlockwise from the positive x-axis. The inverse tangent of y ÷ x gives an acute angle, so use a sketch to see which quadrant the vector is in and adjust.

Two common slips. Adding the entries to find the length: the length of a vector is not x + y, because the movement across and the movement up are at right angles. And taking the angle straight from the calculator for a vector that points left: the display gives an acute angle, and a vector in the second or third quadrant needs 180 added or subtracted.

Worked examples

1. Find the column vector from A(−3, 2) to B(1, 5).

  1. Subtract the coordinates of A from the coordinates of B: across 1 − (−3) = 4 and up 5 − 2 = 3.
  2. The vector is 4
    3    , written AB = (4, 3) in a row when space is short.

2. Find the magnitude of −6
8.

  1. Square the entries and add: 36 + 64 = 100.
  2. Take the square root: 10.

3. Find the direction of −3
4 from the positive x-axis, to one decimal place.

  1. It goes left and up, so it is in the second quadrant. The acute angle with the negative x-axis is tan−1(4 ÷ 3) = 53.1 degrees.
  2. The direction is 180 − 53.1 = 126.9 degrees.

Your go. The first step is given. Find the column vector from P(−2, 3) to Q(4, −1), and its magnitude to one decimal place.

  1. Subtract the coordinates of P from Q: across 4 − (−2), up −1 − 3.
  2. Then use Pythagoras on the two entries and round.
Check the answer

The vector is 6
−4. Its magnitude is √(36 + 16) = √52 = 7.2 to one decimal place.

With a calculator

Check the calculator is in degree mode before using an inverse tangent: the display should show D or DEG. For a magnitude, work out the sum of the squares first and then take the square root, with the whole sum under the root. Round only the final answer.

The key for inverse tangent gives an angle between −90 and 90 degrees only, so it cannot tell a vector in the second quadrant from one in the fourth. Sketch the vector first. For the column (−4, 3), tan−1(3 ÷ 4) gives the acute angle 36.9 degrees from the negative x-axis, and the direction from the positive x-axis is 180 − 36.9 = 143.1.

Check before you go on

1. Which column vector goes from P(5, −1) to Q(−2, 3)?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A boat in a harbour in Penang moves by the vector 9
−12 metres. What is the magnitude of the movement in metres?

3. A hiker walks from P(−1, 2) to Q(4, 14) on a map grid. What is the length of the walk in grid units?

Level 2 · Building

4. The vector x
15 has magnitude 17 and x is positive. What is x?

5. A kite in Guangzhou moves by the vector 4
7 metres in one second. What is its speed in metres per second, to one decimal place?

Level 3 · Stretch

6. A vector is −8
−3. What is its direction, measured anticlockwise from the positive x-axis, to one decimal place?

7. The vector k
k + 1 has magnitude 29 and k is positive. What is k?

Something to think about

The vector (3, 4) and the vector (6, 8) have the same direction but different magnitudes. How are the two columns related, and what is the magnitude of (6, 8) compared with (3, 4)?

One way to think about it

(6, 8) is 2 times (3, 4), so each entry is doubled. Its magnitude is 10, which is also 2 times 5. A vector multiplied by a positive number keeps its direction and its magnitude is multiplied by the same number.

Exit ticket

8. Which vector has the same magnitude as 7
−24?