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Background: Cape sailors use vectors to navigate the stormy seas around the Cape of Good Hope.

Challenge: You’ll be navigators on a treasure hunt!

Objective: Navigate the school field using vectors to find hidden treasure, applying vector addition and direction.

Duration: 20 minutes

Materials Needed (per team):

• Measuring tape

• Compass (for direction)

• Graph paper and pencil

• Calculator

Team Roles:

• Navigator: Uses the compass to find direction.

• Measurer: Measures distances with the tape.

• Recorder: Notes observations and calculations.

Instructions for Students

Step 1: Understand Your Mission (2 minutes)

• Your team has vectors leading to a hidden treasure (marked by a flag).

• Starting point: Center of the rugby field

• Goal: Follow the vectors to find the treasure by navigating and sketching.

Step 2: Plot Vectors on Graph Paper (5 minutes)

• Use graph paper with a scale: 1 cm = 1 m.

• Draw the vectors head-to-tail:

1. Start at (0,0).

2. Use the compass to measure direction (0° = due North, bearing 90 = due East)

3. Draw each vector’s length (magnitude).

• Estimate the treasure’s location by connecting the vectors.

Step 3: Navigate to the Treasure (10 minutes)

• Go to the starting point (rugby field center).

• Follow each vector using the compass (Navigator) and measuring tape (Measurer):

1. Move in the direction and distance of Vector 1.

2. From there, move in the direction and distance of Vector 2, etc.

• Recorder: Note landmarks or observations.

• Find the treasure (flag) and confirm its code with your teacher.

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Group-Specific Challenges

Group 1: Two Vectors to Resultant

Vectors:

1. Vector 1: 6 m at 30° (30° east of north)

2. Vector 2: 4 m at 0° (due north)

Task:

1. Sketch vectors head-to-tail on graph paper (1 cm = 1 m).

2. Navigate to the treasure by following the vectors.

3. Estimate the resultant displacement (magnitude and bearing) by measuring your sketch.

Group 2: Two Vectors to Resultant

Vectors:

1. Vector 1: 5 m at 45° (45° east of north)

2. Vector 2: 3 m at 270° (due west)

Task:

1. Sketch vectors head-to-tail.

2. Navigate to the treasure.

3. Estimate the resultant displacement on your sketch.

Group 3: Three Vectors to Resultant

Vectors:

1. Vector 1: 4 m at 90° (due east)

2. Vector 2: 5 m at 30° (30° east of north)

3. Vector 3: 3 m at 180° (due south)

Task:

1. Sketch vectors head-to-tail.

2. Navigate to the treasure.

3. Estimate the resultant displacement on your sketch.

Group 4: Three Vectors to Resultant

Vectors:

1. Vector 1: 7 m at 315° (45° west of north)

2. Vector 2: 4 m at 0° (due north)

3. Vector 3: 2 m at 270° (due west)

Task:

1. Sketch vectors head-to-tail.

2. Navigate to the treasure.

3. Estimate the resultant displacement on your sketch.

Group 5: Resultant to Component Vectors

Resultant: 7 m at 45° (45° east of north)

Task:

1. Sketch the resultant vector on graph paper.

2. Propose two component vectors that, when added head-to-tail, give the resultant (e.g., one due north, one due east).

3. Navigate to the treasure using the resultant vector.

4. Verify your component vectors by navigating their path to the same treasure.

Hint: Possible components: 4.95 m at 0° (north), 4.95 m at 90° (east).

Calculation and Sketch

Sketch: Draw vectors head-to-tail (Groups 1–4) or resultant and components (Group 5). Label all vectors.

Quick Check: For Groups 1–4, estimate the resultant (magnitude and bearing) from your sketch. For Group 5, list your two component vectors and show they produce the resultant.
[Space for sketch and notes]

Reflection Question

1. How did the head-to-tail method help you navigate? What was the biggest challenge using bearings (0° = north, 90° = east)?

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Definitions
Scalar: A physical quantity with magnitude only.
Vector: A physical quantity with both magnitude and direction.

Watch this video.

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Compared and contrasted, plus examples

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VECTOR DIRECTIONS EXPRESSED IN TWO WAYS:

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VECTOR DIRECTIONS EXPRESSED IN TWO WAYS:

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1. Cardinal directions (NEWS)

2. Bearing (clockwise from North)

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INTRODUCTION TO MECHANICS & WORKING OUT DISTANCE VS DISPLACEMENT

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INTRODUCTION TO MECHANICS & WORKING OUT DISTANCE VS DISPLACEMENT

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As part of our study of Vectors, we need to introduce the following concepts: distance, displacement, speed, velocity and acceleration.

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DISTANCE (d)

Distance:  The magnitude of the actual distance covered by a moving object from the starting point to the finishing point.

DISPLACEMENT (Δx)

Displacement (Δx): The magnitude and direction of the straight-line distance from the starting point to the end point of a motion.

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DISTANCE VS DISPLACEMENT - WATCH THIS VIDEO:

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DISTANCE VS DISPLACEMENT - WATCH THIS VIDEO:

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Definition:
Velocity is the rate of displacement.
Or The rate of change in position.

Remember: "rate" means "per unit time".

Velocity is a vector quantity and is measured in m.s⁻¹.

Average velocity is calculated using the following equation:

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡/𝑡𝑖𝑚𝑒

𝒗𝒂𝒗 = ∆𝒙 / ∆𝑡

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Examples on Distance vs Displacement and Speed vs Velocity

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WITH ANSWERS

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WITH ANSWERS

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DEF:
RESULTANT = the single vector which has the same effect as the original vectors acting together.

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RESULTANT = the single vector which has the same effect as the original vectors acting together.

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RESULTANT = the single vector which has the same effect as the original vectors acting together.

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Click on the link below to do an interactive simulation activity on how to add vectors.

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Remember:
RESULTANT = the single vector which has the same effect as the original vectors acting together.
Calculate the magnitude of the resultant vector, R, in the examples below (use Pythagoras):

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HOW TO DETERMINE THE VECTOR DIRECTION USING SIMPLE TRIG

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HOW TO DETERMINE THE VECTOR DIRECTION USING SIMPLE TRIG

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RESULTANT = the single vector which has the same effect as the original vectors acting together.

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A girl walks a distance of 100 m due east before turning and walking a distance of  50 m in a direction E 60o S.  Determine her resultant displacement by accurate construction and measurement.
Remember:
RESULTANT = the single vector which has the same effect as the original vectors acting together.

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HEAD-TO-TAIL METHOD

TAIL-TO-TAIL METHOD

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1. ADDING VECTORS USING THE TRIANGLE METHOD 

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1. ADDING VECTORS USING THE TRIANGLE METHOD 

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HEAD-TO-TAIL METHOD

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TAIL-TO-TAIL METHOD

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Three vectors of 3 N, 60o, 4 N, 90o and 5 N, 230o act at a point.  Determine the resultant by accurate construction and measurement.

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It means:
Breaking up a skew vector into horizontal and vertical dimensions (lengths).
Horizontal component =  R x
Vertical component = R y
Do the example below:

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MEASURING MOTION IN ONE DIRECTION

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CTSC practical experiment: Investigating motion in one direction