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B. Kinematics · Kinematics: Motion in One and Two Dimensions

Vectors in Two Dimensions: Directions and Scale Diagrams

B2.5 B3.2 A1.12About 70 minutes
Learning goals
  • I can write and draw two-dimensional directions such as [N 30° E] and [E 30° N].
  • I can add vectors tip to tail and find the resultant with a scale diagram.
  • I can find the resultant of two perpendicular vectors with the Pythagorean theorem and tangent.
  • I can subtract vectors by adding the opposite vector, and use this to find displacement in two dimensions.

Words to know

resultant
The single vector that has the same effect as two or more vectors added together.
tip-to-tail method
Adding vectors by drawing each new vector starting at the tip (arrowhead) of the one before. The resultant goes from the first tail to the last tip.
scale diagram
An accurate drawing in which a chosen length represents a quantity, such as 1 cm = 0.5 km.
[N 30° E]
A direction: start facing north and turn 30° toward east.
opposite vector (−v)
A vector with the same magnitude as v but the opposite direction.

Life is not a straight line

A canoe crosses a lake, a hiker follows a trail around a hill, a snowmobile cuts across a field. Motion in the real world happens on a flat map (two dimensions), not just along a line. In one dimension, + and − were enough to show direction. In two dimensions we need angles, and we need a way to add vectors that point in different directions.

Writing directions

In this course, directions in a plane are written in square brackets using the compass points:

[N 30° E] means: start facing north, then turn 30° toward east.

The first letter is where you start; the last letter is the way you turn. The same direction can be written two ways, because the angles from north and from east add to 90°: [N 30° E] = [E 60° N]. Halfway directions can be shortened: [NE] means [N 45° E].

Pilots and sailors also use bearings: an angle measured clockwise from north, written as three digits. [N 30° E] is a bearing of 030°, and [S 20° W] is 200°.

NSEW30°[N 30° E]: start at N, turn 30° toward ENSEW30°[E 30° N]: start at E, turn 30° toward N
[N 30° E] and [E 30° N] are different directions. Each starts from the first compass point named and turns 30° toward the second. Note that [E 30° N] is the same as [N 60° E].

Adding vectors: tip to tail

To add vectors, draw them tip to tail:

  1. Draw the first vector to scale, starting at the origin.
  2. Start the next vector at the tip of the first one, keeping its own length and direction.
  3. Continue for every vector.
  4. The resultant runs from the tail of the first vector to the tip of the last. Measure its length (convert with the scale) and its angle (with a protractor).

The order does not matter: A + B gives the same resultant as B + A. In one dimension this method is exactly what you did on the number line in Lesson 1.

Worked example

Example 1: Two perpendicular displacements

A hiker walks 3.0 km [E] and then 4.0 km [N]. Find her resultant displacement.

  1. Draw 3.0 km [E], then from its tip draw 4.0 km [N]. The resultant joins start to finish and forms the hypotenuse of a right triangle.
  2. Magnitude (Pythagorean theorem): |Δd| = √(3.0² + 4.0²) = √25 = 5.0 km.
  3. Direction: the angle at the start, measured from east toward north, is θ = tan⁻¹(opposite ÷ adjacent) = tan⁻¹(4.0 ÷ 3.0) ≈ 53°.
  4. So Δd = 5.0 km [E 53° N], which is the same as [N 37° E].

5.0 km [E 53° N] (or [N 37° E])

Worked example

Example 2: A scale diagram

A canoeist paddles 2.0 km [N], then 3.0 km [N 60° E]. Use a scale diagram to find her resultant displacement.

  1. Choose a scale that fits your page, for example 1 cm = 0.5 km. Then 2.0 km → 4.0 cm and 3.0 km → 6.0 cm.
  2. Draw a north line. Draw the first vector 4.0 cm straight up (north).
  3. At its tip, draw a faint north line. Use a protractor to measure 60° from north toward east, and draw the second vector 6.0 cm long in that direction.
  4. Draw the resultant from the starting point to the final tip. Measure it: about 8.7 cm, which is 8.7 × 0.5 ≈ 4.4 km.
  5. Measure the angle between the north line at the start and the resultant: about 37° toward east.

About 4.4 km [N 37° E]

2.0 km [N]3.0 km [N 60° E]resultant ≈ 4.4 km[N 37° E]60°37°start1.0 kmN
Example 2 drawn to scale (1.0 km per bar length). Drawing the vectors tip to tail and measuring the dashed resultant gives about 4.4 km [N 37° E]. In the next lesson you will check this with components.

Subtracting vectors

The opposite of a vector, −v, has the same magnitude but points the other way. If v = 5 m [N 20° E], then −v = 5 m [S 20° W].

To subtract, add the opposite: A − B = A + (−B).

This matters because displacement is a subtraction: Δd = d₂ − d₁ = d₂ + (−d₁). In two dimensions you cannot just subtract the numbers; you must flip d₁ and add tip to tail.

Worked example

Example 3: Displacement from two positions

Relative to the school's front door, Ravi's starting position is 400 m [W] and his final position is 300 m [N]. Find his displacement.

  1. Δd = d₂ − d₁ = d₂ + (−d₁) = 300 m [N] + 400 m [E] (the opposite of 400 m [W]).
  2. These are perpendicular: |Δd| = √(300² + 400²) = 500 m.
  3. Angle from north toward east: tan⁻¹(400 ÷ 300) ≈ 53°.
  4. Δd = 500 m [N 53° E]. Check by picture: going from a point west of the door to a point north of it means moving north and east. ✓

500 m [N 53° E]

Drawing good vector diagrams (A1.12)

  • Always draw a north arrow (or label + directions) and state your scale.
  • Use a sharp pencil, a ruler and a protractor; measure angles from the correct reference line (the first letter of the direction).
  • Make the diagram big: larger drawings give more precise answers.
  • Label every vector with its magnitude and direction, and mark the resultant clearly (a different line style helps).
  • Scale diagrams are usually good to about 2 significant digits. The algebraic method in the next lesson is more precise.
Just for fun

Ask for directions in Canada and you might get three kinds of answers:

  • Scalar: "It's about 2 km."
  • Vector: "It's 2 km [N 30° E]."
  • Local: "Turn left where the big maple tree used to be."
Hands-on activity

Walk a vector sum

You need: a measuring tape, a phone compass app (or a magnetic compass), chalk or two small markers, a partner, and an open space such as a yard or park.

  1. Mark a start point. Use the compass to face north. Walk 8.0 m [N] and mark the spot.
  2. From there, face east and walk 6.0 m [E]. Mark the finish.
  3. Measure the straight-line distance from start to finish, and use the compass to find its direction.
  4. Now draw both trips as a scale diagram on paper (for example 1 cm = 1 m) and measure the resultant.
  5. Calculate it with the Pythagorean theorem and tan⁻¹. Compare all three answers.
  6. Try a trip that is not at right angles, such as 8.0 m [N] then 6.0 m [N 45° E], and use only the scale diagram and the tape.

What to look for: the first trip should give 10.0 m [N 37° E] (since √(64 + 36) = 10, tan⁻¹(6 ÷ 8) ≈ 37°). Measured results within about 0.3 m and 5° are good. (Phone compasses can be thrown off by nearby metal and by the difference between magnetic north and true north, so expect a few degrees of difference.)

Your turn

1

Write [N 25° W] in the other form.

2

What direction is [SE] written with an angle?

3

A dog runs 12 m [S] then 5.0 m [W]. Find its resultant displacement.

4

What is the opposite of 7.0 m/s [N 40° W]?

5

Position 1 is 60 m [E] of a tree; position 2 is 80 m [S] of it. Find the displacement from position 1 to position 2.

6

Does the order of adding vectors change the resultant? Explain with an example.

More practice: warm-up

1

In [E 20° S], which way do you face first?

2

Convert [N 30° E] to a bearing.

3

In the tip-to-tail method, where does the resultant start and end?

4

With the scale 1 cm = 5 m, how long is the arrow for 35 m?

5

Add 6 m [E] and 8 m [N]: magnitude only.

More practice: on your own

1

A plane flies 300 km [W] then 400 km [S]. Find its displacement.

2

A boat travels 5.0 km [N] and then 5.0 km [E]. Find its displacement.

3

Using a scale diagram, add 4.0 m [E] and 3.0 m [N 30° E]. (Then check with the answer: about 6.1 m [E 25° N].)

4

A soccer ball is at 20 m [N] of the centre spot, then at 15 m [W] of it. Find its displacement.

5

Two vectors of 5 m and 3 m are added. What are the largest and smallest possible magnitudes of the resultant?

Challenge

1

A hiker walks 2.0 km [N], 3.0 km [E] and 6.0 km [S]. Find her resultant displacement.

2

Explain why the magnitude of A − B can be larger than the magnitude of A + B. Give an example.

3

A scale diagram uses 1 cm = 20 km. Your ruler measures to ±0.1 cm and your protractor to ±1°. How precise is a resultant of 13.4 cm, and why might the algebraic method be preferred?

Check your understanding

Quiz

1.

[N 30° E] means:

2.

[N 30° E] is the same direction as:

3.

In tip-to-tail addition, the resultant is drawn from:

4.

A cyclist rides 9.0 km [W] and then 12 km [N]. The magnitude of her displacement is:

5.

To find A − B, you:

6.

The opposite of 4.0 m [S 15° E] is:

For parents and teachers

Big idea: Expectation B2.5 asks students to solve displacement problems with vector diagrams; this lesson does it with scale diagrams and, for right angles, the Pythagorean theorem and tangent (Grade 10 trigonometry). It also reinforces scalars vs vectors (B3.2) and careful vector diagrams (A1.12).

How to help: Check that every diagram has a north arrow and a scale. Ask your teen to say a direction out loud: "[S 20° W] means face south, turn 20 degrees toward west."

Common difficulties:

  • Measuring the angle from the wrong axis ([N 30° E] vs [E 30° N]).
  • Drawing the second vector from the origin instead of from the tip of the first.
  • Subtracting magnitudes instead of adding the opposite vector.
  • Calculator in radian mode (answers like 0.93 instead of 53°).