C. Geometry and Algebra of Vectors · Vectors in Two-Space and Three-Space

Drawing Vectors: Directed Line Segments, Bearings and Equal Vectors

C1.2About 70 minutes

Learning goals

I can draw a vector in two-space as a directed line segment using a scale.
I can describe a direction as a standard angle, a true bearing (such as 320°) or a quadrant bearing (such as N 40° W), and convert between them.
I can recognize equal, opposite and parallel vectors and the zero vector.
I can name equal vectors in parallelograms and other figures.

Words to know

true bearing
A direction measured clockwise from north, written with three digits (for example 040°).

quadrant bearing
A direction given from north or south toward east or west, such as N 40° W.

standard angle
A direction measured anticlockwise from the positive x-axis, as for an angle in standard position.

equal vectors
Vectors with the same magnitude and the same direction, wherever they are drawn.

opposite vector
The vector −𝐮, with the same magnitude as 𝐮 and the opposite direction.

zero vector
The vector 𝟎 with magnitude 0 and no specific direction.

One arrow, many descriptions

A pilot says "heading three-two-zero". A hiker's guidebook says "N 40° W". A math textbook says "130°". All three can describe exactly the same direction. In this lesson you will learn to read and draw vectors as directed line segments, switch between the common ways of naming a direction, and decide when two arrows represent the same vector.

Three ways to give a direction in two-space

1. Standard (polar) angle θ. Measured anticlockwise from the positive x-axis (east), from 0° up to 360°. This is the same as an angle in standard position, which you used in Grade 11 Functions (MCR3U) and in the Radian Measure and Trigonometric Functions unit of MHF4U.

2. True bearing. Measured clockwise from north, written with three digits: 040°, 135°, 320°. Pilots, sailors and orienteers use these.

3. Quadrant (compass) bearing. Start facing north or south, then turn a stated angle (less than 90°) toward east or west: N 40° W, S 25° E. In SPH3U this was written in square brackets, like [N 40° W], and directions like [W 50° N] (start west, turn 50° toward north) were also allowed; [W 50° N] is the same as N 40° W.

The same vector described three ways: 130° anticlockwise from east, 320° clockwise from north, and 40° west of north.

Quadrant bearing	True bearing β	Standard angle θ
N a° E	a°	90° − a°
S a° E	180° − a°	270° + a°
S a° W	180° + a°	270° − a°
N a° W	360° − a°	90° + a°

Converting between direction systems (0° < a < 90°). In general θ = 90° − β, adding 360° if the result is negative.

Sketch before you convert

Always draw a quick compass cross (N, E, S, W) and put the arrow in the right quadrant first. Then the arithmetic only has to match the picture. The formula θ = 90° − β (add 360° if negative) works both ways: β = 90° − θ (add 360° if negative).

Worked example

Example 1: One direction, three names

A vector has a true bearing of 320°. Write its direction as a quadrant bearing and as a standard angle.

320° clockwise from north is 40° short of a full turn, so the arrow is 360° − 320° = 40° west of north: N 40° W.
Standard angle: θ = 90° − 320° = −230°. Add 360°: θ = 130°.
Check with the sketch: the arrow is in the north-west quadrant (quadrant II), and 130° is between 90° and 180°. ✓

N 40° W, which is the standard angle 130°.

Worked example

Example 2: From a quadrant bearing

Write S 25° E as a true bearing and as a standard angle.

Face south (bearing 180°) and turn 25° toward east. Turning toward east from south is anticlockwise when you look at a map, so the bearing goes down: β = 180° − 25° = 155°.
Standard angle: θ = 90° − 155° = −65°; add 360°: θ = 295°.
Check: south is 270° in standard position, and turning 25° toward east (toward 360°) gives 295°. ✓

True bearing 155°; standard angle 295°.

Drawing a vector as a directed line segment

To draw a vector to scale:

Choose a scale, such as 1 cm : 10 km/h, and write it on the diagram.
Mark the tail and draw a reference line (a north line for bearings, or the positive x-axis for standard angles).
Use a protractor to measure the direction from the reference line.
Draw the arrow with the scaled length and put the arrowhead at the head.
Label the arrow with its name or its magnitude and direction.

Worked example

Example 3: A scale drawing

Draw the velocity 𝐯 = 35 km/h at a true bearing of 240°, using 1 cm : 10 km/h.

Length: 35 ÷ 10 = 3.5 cm.
Direction: 240° is 60° past south (180°), turning toward west, so it is S 60° W.
Draw a north line through the tail. Measure 240° clockwise from it (or 60° from the south direction toward west) and draw a 3.5 cm arrow.

A 3.5 cm arrow pointing S 60° W (bearing 240°), labelled 35 km/h.

Equal, opposite and parallel vectors

A vector is defined only by its magnitude and direction. Its position does not matter. So:

Equal vectors have the same magnitude and the same direction, even if they are drawn in different places. You can slide (translate) a vector anywhere without changing it, as long as you don't turn it or stretch it.
The opposite of 𝐮, written −𝐮, has the same magnitude and the opposite direction. For any points, 𝐁𝐀 = −𝐀𝐁.
Parallel (or collinear) vectors point in the same or opposite directions. Their arrows lie on parallel lines (or on the same line).
The zero vector 𝟎 has magnitude 0 and no particular direction. For example, 𝐀𝐀 = 𝟎.

Vectors that can be moved like this are sometimes called free vectors. A position vector is different: it always starts at the origin, because its job is to locate a point.

All four arrows labelled 𝐮 are equal: each goes 3 right and 2 up. −𝐮 has the same length but points the opposite way. 𝐯 has the same length as 𝐮 but a different direction, so 𝐯 ≠ 𝐮.

Worked example

Example 4: Equal vectors in a parallelogram

ABCD is a parallelogram whose diagonals meet at E. Name a vector equal to each of 𝐀𝐁, 𝐀𝐃 and 𝐀𝐄, and a vector equal to −𝐂𝐁.

Opposite sides of a parallelogram are parallel and equal in length, so 𝐀𝐁 = 𝐃𝐂 (same direction: both point from the left side to the right side) and 𝐀𝐃 = 𝐁𝐂.
The diagonals of a parallelogram bisect each other, so E is the midpoint of AC: 𝐀𝐄 = 𝐄𝐂.
−𝐂𝐁 = 𝐁𝐂, which also equals 𝐀𝐃.

𝐀𝐁 = 𝐃𝐂, 𝐀𝐃 = 𝐁𝐂, 𝐀𝐄 = 𝐄𝐂, and −𝐂𝐁 = 𝐁𝐂 = 𝐀𝐃. Careful: 𝐀𝐁 ≠ 𝐂𝐃, because 𝐂𝐃 points the other way (𝐂𝐃 = −𝐀𝐁).

In parallelogram ABCD, opposite sides give equal vectors when they point the same way. The diagonals bisect each other at E, so 𝐀𝐄 = 𝐄𝐂 and 𝐁𝐄 = 𝐄𝐃.

Just for fun

A pilot and a math teacher can agree on a direction even when one says "three-two-zero" and the other says "one hundred thirty degrees". They just have to agree not to argue about which way is clockwise.

Hands-on activity

Slide, don't spin

You need: graph paper (or lined paper with a ruler), a protractor, a pencil, a small piece of cardboard (a cereal box works), and scissors (use them carefully).

On the graph paper, draw a north arrow at the top. Using 1 cm : 10 km/h, draw these vectors with their tails at different places: 𝐚 = 30 km/h at 060°, 𝐛 = 45 km/h at S 30° W, 𝐜 = 20 km/h at standard angle 180°.
Cut a thin arrow 4 cm long from the cardboard. Lay it on the page along a grid line and slide it to three different places without rotating it. Trace it each time. All three tracings are equal vectors.
Now rotate the cut-out arrow by any angle and trace it. Is this equal to the first tracings? Write a sentence explaining why not.
Convert each direction in step 1 into the other two systems.

What to look for: 𝐚 is 3 cm long at N 60° E (standard angle 30°); 𝐛 is 4.5 cm at bearing 210° (standard angle 240°); 𝐜 is 2 cm pointing due west (bearing 270°). The rotated tracing has the same magnitude but a different direction, so it is not equal.

Your turn

Write the true bearing 075° as a quadrant bearing and as a standard angle.

Write the true bearing 200° as a quadrant bearing and as a standard angle.

Write N 60° W as a true bearing and as a standard angle.

A vector has standard angle 210°. Write it as a true bearing and as a quadrant bearing.

Write the SPH3U-style direction [W 30° S] as a quadrant bearing (starting from N or S).

ABCD is a parallelogram. Is 𝐀𝐁 = 𝐂𝐃? Explain.

Using 1 cm : 50 N, how long is the arrow for a 320 N force? In which direction would you draw it if the force acts at bearing 135°?

More practice: warm-up

What true bearing is due west?

What standard angle is due south?

Write S 40° E as a true bearing.

What is 𝐁𝐀 in terms of 𝐀𝐁?

Two vectors have the same direction and the same magnitude but start at different points. Are they equal?

More practice: on your own

Convert the standard angle 320° to a true bearing and a quadrant bearing.

Convert S 72° W to a true bearing and a standard angle.

A vector has true bearing 0°. Give its standard angle and say which unit vector direction it is.

In a regular hexagon ABCDEF with centre O, name two vectors equal to 𝐀𝐁. (Hint: a regular hexagon is made of six equilateral triangles meeting at O.)

Explain why a position vector, unlike a free vector, cannot be slid to a new location.

Challenge

Find a formula for the true bearing of −𝐮 if 𝐮 has true bearing β. Test it with β = 040° and β = 300°.

The quadrilateral PQRS satisfies 𝐏𝐐 = 𝐒𝐑. Prove that PQRS is a parallelogram.

A ship sails at a bearing of 110°, then turns 75° to port (to its left, which is anticlockwise seen from above). What is its new true bearing, quadrant bearing and standard angle?

Check your understanding

Quiz

Which quadrant bearing matches a true bearing of 230°?

What is the standard angle (anticlockwise from the positive x-axis) of N 30° E?

Which statement about equal vectors is true?

ABCD is a parallelogram (vertices in order). Which vector equals 𝐀𝐁?

A vector has standard angle 300°. What is its true bearing?

Which describes the zero vector?

For parents and teachers

Big idea (C1.2): a 2-D vector can be drawn as an arrow, and its direction can be named in several ways. Vectors with the same magnitude and direction are equal, wherever they are drawn.

How to help: Use a phone compass app together. Point in a direction and ask your teen to name it three ways (true bearing, quadrant bearing, standard angle). A quick compass-cross sketch should come first, every time.

Common difficulties:

Measuring a true bearing anticlockwise, or a standard angle clockwise.
Writing quadrant bearings that start from E or W (in this course they start from N or S).
Thinking 𝐀𝐁 = 𝐂𝐃 in a parallelogram ABCD (it is 𝐃𝐂; 𝐂𝐃 is the opposite).
Forgetting to add 360° when θ = 90° − β comes out negative.