Concept of Direction: Scalars and Vectors

Concept of Direction

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Direction is defined as the line or course along which something moves, faces or lies, measured from a chosen reference point or reference line. For instance, saying “the school is 2 km away” gives only a distance. On the other hand, saying “the school is 2 km north of the market” gives distance and direction, so it fixes the school’s position exactly.

Direction has no meaning on its own. It must always be stated relative to a reference, such as the cardinal points (e.g North), the positive x-axis, the horizontal, or a landmark.

 

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Ways of describing Direction

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1. Everyday words

They include words like up, down, left, right, forward, backward. These depend on where the observer stands, so they are not precise enough for physics.

 

2. Cardinal points (compass points)

The four main directions are North (N), East (E), South (S) and West (W), each 90° from the next. The four intermediate points are North-East (NE), South-East (SE), South-West (SW) and North-West (NW), each lying midway (45°) between two cardinal points.

The table below shows the cardinal points and their respective three figure bearing:

Direction	N	NE	E	SE	S	SW	W	NW
Three-figure bearing	000°	045°	090°	135°	180°	225°	270°	315°

Note: Cardinal points are sometimes called cardinal bearings when expressed as a named direction (such as north or east) rather than in degrees. 

 

3. Compass bearing (quadrant bearing)

A compass bearing is the direction of a point or object measured as an angle relative to a north reference line. In this regard, compass bearing is written as an angle measured from North or South towards East or West. Below are examples of compass bearing:

• N30°E means start facing North and turn 30° towards East.

• S50°W means start facing South and turn 50° towards West.

When writing compass bearing, the first letter is always N or S, and the last letter is always E or W.

 

4. Three-figure bearing

Three-figure bearing is the angle measured clockwise from North, always written with three digits (for example 045°, 090°, 260°). It is the standard method in navigation, aviation and physics problems.

The image below describes a three-figure bearing:

The above image shows a point P which lies on a bearing of 060° (also written N60°E).

Please read on position, distance and displacement here.

 

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Converting Compass Bearings to Three-figure Bearings

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The table below shows the quadrant, rule for determining the bearing, and example of the types of bearing (compass bearing and three-figure bearing).

 

Quadrant	Rule	Compass to Three-figure bearing
N…E (first)	bearing = the angle	N40°E = 040°
S…E (second)	bearing = 180° − angle	S30°E = 150°
S…W (third)	bearing = 180° + angle	S50°W = 230°
N…W (fourth)	bearing = 360° − angle	N60°W = 300°

 

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Back Bearing (Reverse Direction)

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The back bearing is the direction of the return journey. It differs from the forward bearing by 180°.

Example 1

• A ship sails on a bearing of 070°. Its back bearing is 070° + 180° = 250°.

• A plane flying on 300° has a back bearing of 300° − 180° = 120°.

 

Example 2

Find the back bearing of (a) 160° (b) 280° (c) 065°.

• 160°

Since 160° is less than 180°, add 180°:

180° + 160° = 340°

• 280°

Since 288° is greater than 180°, subtract 180°:

288° - 180° = 108°

• 180°

Since it is exactly 180°, subtract 180°

180° - 180° = 0°

The back bearings are 340°, 108°, and 0°.

 

Please read on fundamental, derived and supplementary quantities and units with dimensions here.

 

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Direction on the x–y plane

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In physics diagrams, direction is often given as an angle θ measured anticlockwise from the positive x-axis, or as “θ above the horizontal”. A direction opposite to a chosen positive direction is given a negative sign. For example, if the westward velocity is negative (-5 m/s), and eastward velocity, positive (+5 m/s); the sign tells you the direction along that line. Below is an illustration of this  example.

 

Why direction matters

Two people leave the same point and each walks 5 km. One walks north, the other east. Each has travelled the same distance, yet they end up in different places. Direction is therefore needed to describe motion, forces, fields and many other physical quantities completely.

 

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Scalar Quantities

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A scalar quantity is defined as a physical quantity that has magnitude (size) only. It has no direction.

• A scalar is fully described by a number and a unit, for example 5 kg, 30 s, 25 °C, 100 J.

 

Examples of scalar quantities

Mass, time, distance, speed, temperature, density, volume, area, length, energy, work, power, pressure, electric charge, potential difference and frequency.

 

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Properties of Scalars

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• They are described by magnitude alone.

• They are added, subtracted, multiplied and divided using ordinary arithmetic. For example, 3 kg + 4 kg = 7 kg and 20 °C − 5 °C = 15 °C.

• They can be positive or negative (for example temperature or charge), but the sign does not represent a spatial direction.

Please read on measurement of basic quantities and importance of measurement in physics here.

 

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Vector Quantities

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Examples of vector quantities

Displacement, velocity, acceleration, force, weight, momentum, impulse, electric field strength, gravitational field strength and torque (moment of a force).

 

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Representing a Vector

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Symbolically, a vector is written in bold (A), or with an arrow over the letter (e.g A with → above it). Its magnitude is written |A| or simply A in italics.

Graphically, a vector is drawn as a straight line with an arrow head.

• The length of the line, drawn to a chosen scale, represents the magnitude.

• The arrow head shows the direction.

• The tail is the starting point and the head is the end point.

Choosing a scale: To draw a force of 30 N acting due east with a scale of 1 cm : 10 N, draw a line 3 cm long pointing east and put an arrow head at its end.

• Negative Vector (−A): It has the same magnitude as A but opposite direction.

• Unit Vector: This is a vector of magnitude (length) of exactly 1, used only to show direction. A unit vector  represents pure direction with no scaling or units attached. The unit vectors along the x, y and z axes are written i, j and k. (i points along the positive x-axis (1, 0, 0), j points along the positive y-axis 0, 1, 0), and k points along the positive z-axis (0, 0, 1).

• Resultant Vector: This is a single vector that has the same effect as two or more vectors acting together.

You can attempt various questions in physics here.

 

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Adding Vectors

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Vectors cannot be added by simple arithmetic because the direction matters. Two vectors of magnitude 3 units and 4 units can give a resultant anywhere between 1 and 7 units, depending on the angle between them.

The table below shows the situation surrounding vectors (A and B)  and their resultant magnitude.

Situation	Resultant magnitude
Same direction (angle 0°)	A + B (the maximum)
Opposite directions (angle 180°)	A − B (the minimum), in the direction of the larger
At right angles (90°)	√(A² + B²) by Pythagoras’ theorem, at angle tan⁻¹(B/A) to A

 

Note: For other angles, use the triangle law (place the tail of the second vector at the head of the first; the resultant joins the first tail to the last head) or the parallelogram law.

 

Worked Example 1

A girl walks 6 m east and then 8 m north. Find her distance travelled and her displacement.

Solution

This question presents us with a right angle situation since the girl walks 6 m east and 8 m north. This is shown via the diagram below.

• Distance = 6 + 8 = 14 m (scalar, ordinary addition).

• Displacement magnitude: Going by the question, the displacement is solved using pythagoras.

Displacement = √(6² + 8²) = √100 = 10 m.

• Direction: tan θ = B/A

tan θ = 8/6 = 1.33

θ = 53.1° north of east, which is a bearing of 90° − 53.1° ≈ 037°.

The image below intepretes the question and solution above: 6 m east, 8 m north 10 m (resultant).

The resultant displacement (10 m) is shorter than the total distance (14 m) walked.

You can read fun facts on gravity here.

 

Worked Example 2

A boy runs 100 m east in 20 s and then 100 m west in 30 s along a straight road. Calculate (a) his average speed and (b) his average velocity for the whole journey.

Solution

• Average Speed = Total Distance/Total Time

Average speed depends on the total distance traveled regardless of direction.

Total Distance: 100 m (east) + 100 m (west) = 200 m

Total Time: 20 s + 30 s = 50 s

Average Speed = 200 m/50 s = 4 m/s

 

• Average Velocity = Net Displacement/Total Time

Average velocity depends on the net displacement, which takes direction into account. If we choose East as the positive direction (+) and West as the negative direction (-):

Net Displacement: +100 m + (-100 m) = 0 m. (This implies that the boy ends up exactly where he started)

Total Time = 50 s

Average Velocity = 0/50 = 0 m/s

 

Worked Example 3

A ship sails 30 km on a bearing of 090° and then 40 km on a bearing of 000°. Find how far it is from its starting point and the bearing of the starting point from the ship (its back bearing).

Solution

Below is an image interpretation of this question.

• Going by the question, finding the distance from the starting point can be solved using Pythagoras theorem (a² + b² = c²). This is so because the ship's journey forms a right-angled triangle (at 90°).

First leg: 30 km on a bearing of 090° (due East). Second leg: 40 km on a bearing of 000° (due North).

Using the Pythagorean theorem:

Distance² = Eastward Distance² +bNorthward Distance²

Distance² = 30² + 40²

Distance² = 900 + 1600 =2500

Distance = √2500

Distance = 50 km

 

• Bearing of the ship from the start

To find the bearing of the ship from the starting point (back bearing), we calculate the angle inside the triangle from the true North line:

Tan θ = Opposite (East)/Adjacent (North) = 30/40 = 0.75

θ = arctan (0.75) ≈ 36.87°

The forward bearing from the starting point to the final position is 036.9°.

You can read on gravity (on the solar system) here.

 

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Differences between Scalars and Vectors

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The table below shows the comparison (differences) between a scalar and Vector.

Scalar	Vector
Magnitude only	Magnitude and direction
Changes only if the magnitude changes	Changes if the magnitude or the direction changes
Added by ordinary arithmetic	Added using the triangle or parallelogram law, or by components
Cannot be represented by an arrow	Represented by an arrow
Examples: mass, time, speed, energy	Examples: force, velocity, momentum, weight

 

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Related Scalar–Vector Pairs

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Practice Questions

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Section A: Objective questions

Choose the correct option. Answers are at the end of this section.

1. Which of the following is a vector quantity? A. Speed B. Mass C. Displacement D. Time

2. Which of these is a scalar quantity? A. Velocity B. Acceleration C. Energy D. Weight

3. The three-figure bearing of due West is _____. A. 090° B. 180° C. 270° D. 360°

4. S30°E expressed as a three-figure bearing is _____. A. 030° B. 120° C. 150° D. 210°

5. The back bearing of 065° is _____. A. 115° B. 155° C. 245° D. 295°

6. A boy runs once round a circular track and returns to his starting point. His displacement is _____. A. equal to the circumference B. equal to the diameter C. equal to the radius D. zero

7. Which pair contains only vector quantities?A. Mass and speed B. Weight and velocity C. Work and energy D. Time and temperature

8. Two forces of 3 N and 4 N act on a body. Which of these cannot be their resultant? A. 1 N B. 5 N C. 7 N D. 8 N

9. A vector is represented graphically by _____. A. a dot B. a directed line segment C. a curved line D. a number only

10. A man walks 4 m east and then 3 m west. His displacement is _____. A. 1 m east B. 1 m west C. 7 m east D. 7 m west

11. A vector of magnitude one is called _____. A. a null vector B. a resultant C. a unit vector D. a scalar

12. A car travels at a constant speed of 20 m/s round a roundabout. Which statement is correct? A. Its velocity is constant B. Its velocity changes because its direction changes C. Its speed changes D. Its displacement is constant

 

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Section B: Theory questions

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1. (a) Define direction. (b) Explain why a reference is needed when stating a direction.

2. Define a scalar quantity and a vector quantity. State five examples of each.

3. Explain the difference between (a) distance and displacement, and (b) speed and velocity.

4. Describe how a vector is represented (a) in symbols and (b) graphically. How would you draw a force of 50 N acting due north using a scale of 1 cm : 10 N?

5. Explain what is meant by a three-figure bearing and a back bearing. Give one example of each.

6. Why can the sum of two vectors of magnitude 5 N and 12 N be anything from 7 N to 17 N, while the sum of two masses of 5 kg and 12 kg is always 17 kg?

7. A body moves in a circle at constant speed. Explain why its velocity is not constant.

8. State the differences between scalars and vectors in a table with at least four rows.

 

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Section C: Calculations

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1. Convert each compass bearing to a three-figure bearing: (a) N25°E (b) S40°E (c) S65°W (d) N10°W.

2. Find the back bearing of (a) 130° (b) 300° (c) 045°.

3. A student walks 9 m east and then 12 m north. Calculate (a) the distance travelled, (b) the magnitude of the displacement, and (c) the direction of the displacement as a three-figure bearing, correct to the nearest degree.

4. Two forces of 6 N and 8 N act at a point. Find the magnitude of the resultant when the forces act (a) in the same direction, (b) in opposite directions, (c) at right angles to each other.

5. A runner completes one full lap of a circular track of radius 70 m. Calculate the distance covered and the displacement. Repeat for half a lap. (Take π = 22/7.)

6. A car travels 180 km due north in 3 hours. Find (a) its average speed and (b) its average velocity.

7. A boy runs 100 m east in 20 s and then 100 m west in 30 s along a straight road. Calculate (a) his average speed and (b) his average velocity for the whole journey.

8. A ball is thrown vertically upward and rises 20 m before falling back to the thrower’s hand. State the distance travelled and the displacement.

9. A ship sails 30 km on a bearing of 090° and then 40 km on a bearing of 000°. Find how far it is from its starting point and the bearing of the starting point from the ship (its back bearing).

 

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Section D: Challenge questions

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1. Can the magnitude of the resultant of two vectors be smaller than the magnitude of either vector? Give a numerical example.

2. Is it possible for the distance travelled to be zero while the displacement is not zero? Explain.

3. Electric current has both magnitude and a direction of flow, but it is classified as a scalar. Suggest a reason.