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Chapter 10 · Vector Algebra

Exercise 10.1 · Q2: Scalar or vector: 10 kg, 2 m north-west, 40°, 40 watt, 10−1910^{-19} coulomb, 20 m/s²

Exercise 10.1Scalars and vectors · Sections 10.1 and 10.2
The question

Classify the following measures as scalars and vectors.

(i) 10 kg    (ii) 2 meters north-west    (iii) 40°

(iv) 40 watt    (v) 10−1910^{-19} coulomb    (vi) 20 m/s2^2

Exam answer

(i) 10 kg measures mass, which has only magnitude: scalar.

(ii) 2 meters north-west is a displacement, with magnitude 2 m and a direction: vector.

(iii) 40° measures an angle, which has only magnitude: scalar.

(iv) 40 watt measures power, which has only magnitude: scalar.

(v) 10−1910^{-19} coulomb measures electric charge, which has only magnitude: scalar.

(vi) 20 m/s² measures acceleration, which has magnitude and direction: vector.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

A scalar is a quantity that involves only one value, its magnitude, a real number. A vector is a quantity that has magnitude as well as direction (Section 10.1 and Definition 1). So for each measure we ask: what physical quantity does its unit measure, and does that quantity need a direction to be complete? As in the book's Example 2 (where "10 Newton" is classed as force, a vector), we classify the quantity the measure belongs to.

Given, to find, and the plan
Given
Six measures: 10 kg; 2 meters north-west; 40°; 40 watt; 10−1910^{-19} coulomb; 20 m/s2^2⁠.
To find
Whether each is a scalar or a vector.
Plan
For each measure, (1) name the quantity its unit measures, (2) decide whether that quantity has a direction, (3) classify.

Basics you need

1Scalars and vectors

Scalar: one value only, the magnitude (examples in Section 10.1: length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density, resistance). Vector: magnitude and direction (examples: displacement, velocity, acceleration, force, weight, momentum, electric field intensity). Basics: some answers need a "which way"

2Units tell you the quantity

kg is a unit of mass, ° a unit of angle, watt a unit of power, coulomb a unit of electric charge, m/s2^2 a unit of acceleration, and metres (with a compass direction) a displacement.

3How the book classifies

Example 2: "10 Newton" is Force, a vector, although no direction is printed; "30 km/hr" is Speed, a scalar; "20 m/s towards north" is Velocity, a vector. The classification goes by the quantity. Lesson: Examples 1–3

Part (i): 10 kg

  1. Name the quantity and classify

    The kilogram is the unit of mass. A mass of 10 kg is completely described by the number 10 and the unit; "10 kg towards the east" makes no sense.

    So 10 kg is a scalar.

    A quantity is a scalar when one real number (with its unit) says everything about it. Two masses always combine by plain addition: 10 kg and 10 kg make 20 kg, whatever their positions.
    Mass is in the book's list of scalar quantities (Section 10.1). Basics: some answers need a "which way"

Part (ii): 2 meters north-west

  1. Name the quantity

    "2 meters north-west" tells us how far (2 m) and which way (north-west, halfway between north and west, 45° from each). A length together with a direction is a displacement.

    N S E W Q O 45° 2 m scalar: 10 kg one number, no arrow vector: 2 m north-west a length and a way
    "2 meters north-west" drawn as the directed line segment OQ→\overrightarrow{OQ}⁠: length 2 m (the tick marks 1 m) at 45° from north towards west. A mass such as 10 kg cannot be drawn as an arrow: it has no direction.
    The phrase gives both a magnitude and a direction, which is exactly Definition 1 of a vector. Without "north-west", 2 meters would only be a length (a scalar); the compass direction is what turns it into a displacement.
    Displacement is in the book's list of vector quantities (Section 10.1). Lesson: a vector is a length with a direction

  2. Classify

    Magnitude 2 m and direction north-west: 2 meters north-west is a vector.

    The direction really matters: 2 m north-west followed by 2 m south-east brings you back to the start (total displacement 0), while 2 m north-west twice takes you 4 m north-west. A scalar could not behave like this.
    A quantity that has magnitude as well as direction is called a vector (Definition 1, Section 10.2). Basics: some answers need a "which way"

Part (iii): 40°

  1. Name the quantity and classify

    40° is an angle: it measures how much one line is turned from another. The number 40 with the unit "degree" describes it completely; it does not point anywhere by itself.

    So 40° is a scalar.

    Angles are used to describe directions (as in "30° east of north"), but an angle on its own is just a size of turning, one real number. To make a direction you also need a starting line and a side to turn to; to make a vector you also need a magnitude.
    A scalar involves only one value, its magnitude (Section 10.1). Basics: some answers need a "which way"

Part (iv): 40 watt

  1. Name the quantity and classify

    The watt is the unit of power, the rate of doing work (1 watt = 1 joule of work per second). A 40 watt bulb uses energy at the rate 40 joules per second; there is no direction in that statement.

    So 40 watt is a scalar.

    Work is a scalar (Section 10.1 lists it among the scalars), and time is a scalar, so their ratio, power, is a scalar too: dividing one real number by another gives a real number, not an arrow.
    Work and time are scalar quantities (Section 10.1). Basics: some answers need a "which way"

Part (v): 10−1910^{-19} coulomb

  1. Name the quantity and classify

    The coulomb is the unit of electric charge. A charge of 10−1910^{-19} coulomb (a very small charge; one electron carries about 1.6×10−191.6 \times 10^{-19} coulomb) is described by one number and its unit.

    So 10−1910^{-19} coulomb is a scalar.

    The minus sign is in the exponent: 10−19=1101910^{-19} = \frac{1}{10^{19}} is a small positive number, not a direction. Even charges that are negative are scalars: the sign of a charge is part of the real number, and it does not point anywhere in space.
    A scalar is a quantity given by one real number (Section 10.1). Basics: some answers need a "which way"

Part (vi): 20 m/s²

  1. Name the quantity

    The unit m/s2^2 (metres per second, per second) is the unit of acceleration, the rate of change of velocity. Velocity is a vector, so its rate of change has a direction too: a car braking and a car speeding up can have accelerations of the same size in opposite directions.

    No direction is printed with "20 m/s2^2", exactly as none is printed with "10 Newton" in Example 2, which the book classes as force, a vector. The measure gives the size of an acceleration; to state that acceleration completely you must also say which way it acts.
    Acceleration and velocity are in the book's list of vector quantities (Section 10.1). Lesson: Examples 1–3

  2. Classify

    Acceleration has magnitude and direction: 20 m/s2^2 is a vector (a measure of a vector quantity).

    This matches the book's rule in Example 2: the measure is classified by the quantity it belongs to. Compare "30 km/hr", which is a speed (a scalar), with "20 m/s towards north", which is a velocity (a vector).
    A quantity that has magnitude as well as direction is a vector (Definition 1, Section 10.2). Basics: some answers need a "which way"
Result

(i) 10 kg (mass): scalar. (ii) 2 meters north-west (displacement): vector. (iii) 40° (angle): scalar. (iv) 40 watt (power): scalar. (v) 10−1910^{-19} coulomb (charge): scalar. (vi) 20 m/s2^2 (acceleration): vector.

(Textbook answer: (i) scalar (ii) vector (iii) scalar (iv) scalar (v) scalar (vi) vector.)

Check it with numbers

Scalars add as plain numbers: 10 kg and 10 kg always make 20 kg. Vectors do not. Walk 2 m north-west and then 2 m north-east. North-west has east and north parts (−2,2)\left(-\sqrt2, \sqrt2\right) m and north-east has (2,2)\left(\sqrt2, \sqrt2\right) m, so the total is (0,22)\left(0, 2\sqrt2\right)⁠: you end up 22≈2.8282\sqrt2 \approx 2.828 m due north of the start, not 4 m. The answer depends on the directions, which is what makes displacement a vector.

Common mistakes
  1. Calling 40° a vector because angles describe directions. An angle alone is a single number; a direction needs a reference line and a side, and a vector needs a magnitude as well.
  2. Reading the minus sign in 10−1910^{-19} as a direction. It is part of the exponent: 10−1910^{-19} is a small positive number.
  3. Calling 20 m/s2^2 a scalar because no direction is printed. The unit shows it is an acceleration, a vector quantity, just as "10 Newton" in Example 2 is a force.
  4. Confusing power with force. Power (watt) is energy per second, a scalar; force (newton) is a vector.
What to take away
  1. Classify a measure by the quantity its unit measures: mass, angle, power, charge, time, distance, speed and work are scalars; displacement, velocity, acceleration and force are vectors.
  2. A test: can the quantity sensibly be followed by "towards the north"? If it must be, it is a vector.
  3. Signs and exponents are part of the number, not directions.

Every fact used, and where it comes from

FactUsed inSource
Scalar: one value (magnitude), a real number; the lists of scalar and vector quantitiesParts (i) to (vi)Section 10.1 (Basics: some answers need a "which way")
A quantity with magnitude and direction is a vectorParts (ii), (vi)Definition 1, Section 10.2
Classification by quantity: force (vector), speed (scalar), velocity (vector)Idea; Part (vi)Example 2, Section 10.3 (Lesson: Examples 1–3)
Units: kg (mass), degree (angle), watt = joule per second (power), coulomb (charge), m/s2^2 (acceleration)Parts (i) to (vi)Physics units (SI)
10−19=1101910^{-19} = \frac{1}{10^{19}}Part (v)Laws of exponents
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