Class 12 Maths, Worked in Full Past papersGet updates
Chapter 10 · Vector Algebra

Example 2: Scalar or vector: 5 seconds, 1000 cm³, 10 Newton, 30 km/hr, 10 g/cm³, 20 m/s north

ExamplesScalars and vectors · Sections 10.1 and 10.2
The question

Classify the following measures as scalars and vectors.

(i) 5 seconds    (ii) 1000 cm3^3    (iii) 10 Newton

(iv) 30 km/hr    (v) 10 g/cm3^3    (vi) 20 m/s towards north

Exam answer

(i) 5 seconds measures time, which has only magnitude: scalar.

(ii) 1000 cm³ measures volume, which has only magnitude: scalar.

(iii) 10 Newton measures force, which has magnitude and direction: vector.

(iv) 30 km/hr measures speed, which has only magnitude: scalar.

(v) 10 g/cm³ measures density, which has only magnitude: scalar.

(vi) 20 m/s towards north is a velocity, with magnitude 20 m/s and a 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: which physical quantity does its unit measure, and does that quantity need a direction to be complete? We classify the quantity: "10 Newton" names a force even though no direction is printed, and a force always acts in some direction.

Given, to find, and the plan
Given
Six measures: 5 seconds; 1000 cm3^3⁠; 10 Newton; 30 km/hr; 10 g/cm3^3⁠; 20 m/s towards north.
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 (Section 10.1 lists length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density, resistance). Vector: magnitude and direction (displacement, velocity, acceleration, force, weight, momentum, electric field intensity). Basics: some answers need a "which way"

2Units tell you the quantity

The second is a unit of time, cm3^3 of volume, the newton of force, km/hr of speed (distance per time), g/cm3^3 of density (mass per volume), and m/s with a compass direction of velocity.

3Speed and velocity

Speed says only how fast; velocity says how fast and which way. A measure such as "30 km/hr" with no direction is a speed; "20 m/s towards north" is a velocity. Lesson: Examples 1–3

Part (i): 5 seconds

  1. Name the quantity and classify

    The second is the unit of time. "5 seconds" is completely described by the number 5 and the unit; "5 seconds towards the east" makes no sense.

    So 5 seconds is a scalar (time).

    A quantity is a scalar when one real number (with its unit) says everything about it. Two time intervals always combine by plain addition: 5 s and 5 s make 10 s.
    Time is in the book's list of scalar quantities (Section 10.1). Basics: some answers need a "which way"

Part (ii): 1000 cm³

  1. Name the quantity and classify

    The cubic centimetre is a unit of volume: 1000 cm3^3 is the space taken up by a litre of water. A volume has a size but does not point anywhere.

    So 1000 cm3^3 is a scalar (volume).

    The "3" is an exponent of the unit (centimetre cubed), not a direction. Volumes add like ordinary numbers: two 1000 cm3^3 bottles hold 2000 cm3^3⁠, whatever way they face.
    Volume is in the book's list of scalar quantities (Section 10.1). Basics: some answers need a "which way"

Part (iii): 10 Newton

  1. Name the quantity and classify

    The newton is the unit of force. A force is a push or a pull, and every push or pull acts in some direction: 10 N pushing a box to the left and 10 N pushing it to the right have opposite effects.

    So 10 Newton is a vector (force).

    The measure "10 Newton" prints only the magnitude of the force, but the quantity it measures needs a direction to be complete. We classify the quantity, and force is a vector; the book's own answer is "Force-vector".
    Force is in the book's list of vector quantities (Section 10.1); a vector has magnitude as well as direction (Definition 1, Section 10.2). Basics: some answers need a "which way"

Part (iv): 30 km/hr

  1. Name the quantity and classify

    km/hr is distance per time, a unit of rate of motion. No direction is given, so "30 km/hr" says only how fast something moves: it is a speed.

    So 30 km/hr is a scalar (speed).

    Speed is the size of a velocity with the direction left out. A car reading 30 km/hr on its speedometer could be going north, east or round a bend; the reading is the same number. Compare Part (vi), where the same kind of unit comes with a direction.
    Speed is in the book's list of scalar quantities (Section 10.1). Basics: some answers need a "which way"

Part (v): 10 g/cm³

  1. Name the quantity and classify

    g/cm3^3 is mass per volume, the unit of density. A density of 10 g/cm3^3 says how much mass is packed into each cubic centimetre; there is no direction in that statement.

    So 10 g/cm3^3 is a scalar (density).

    Mass is a scalar and volume is a scalar, so their ratio is one real number: dividing a number by a number gives a number, not an arrow.
    Density is in the book's list of scalar quantities (Section 10.1). Basics: some answers need a "which way"

Part (vi): 20 m/s towards north

  1. Name the quantity

    "20 m/s towards north" says how fast (20 m/s) and which way (north). A rate of motion together with a direction is a velocity.

    start N 20 m/s towards north Scale 5 m/s 30 km/hr speed: how fast only no arrow, no direction velocity: a vector
    Left: "20 m/s towards north" drawn as an arrow, 4 scale lengths of 5 m/s, pointing north: magnitude and direction, a vector. Right: "30 km/hr" is a reading on a dial, how fast only, with nothing to say which way: a scalar.
    The phrase gives both a magnitude and a direction, which is exactly Definition 1 of a vector. Without "towards north", 20 m/s would only be a speed (a scalar); the direction is what turns it into a velocity.
    Velocity is in the book's list of vector quantities (Section 10.1). Lesson: a vector is a length with a direction

  2. Classify

    Magnitude 20 m/s and direction north: 20 m/s towards north is a vector (velocity).

    The direction really matters: a boat moving 20 m/s north on a river that carries it 20 m/s south does not move at all, while a boat moving 20 m/s north on water that also flows 20 m/s north moves at 40 m/s north. Velocities combine by the triangle law, not by plain addition, which is the mark of a vector.
    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"
Result

(i) 5 seconds: time, scalar. (ii) 1000 cm3^3⁠: volume, scalar. (iii) 10 Newton: force, vector. (iv) 30 km/hr: speed, scalar. (v) 10 g/cm3^3⁠: density, scalar. (vi) 20 m/s towards north: velocity, vector.

(Textbook answer: (i) Time-scalar (ii) Volume-scalar (iii) Force-vector (iv) Speed-scalar (v) Density-scalar (vi) Velocity-vector.)

Check it with numbers

Scalars add as plain numbers: 5 s + 5 s = 10 s, and 1000 cm3^3 + 1000 cm3^3 = 2000 cm3^3⁠. Vectors do not: 20 m/s north together with 20 m/s south gives 0 m/s, not 40 m/s.

Speed and velocity can use the same units: 30 km/hr =30×10003600= \frac{30 \times 1000}{3600} m/s =253= \frac{25}{3} m/s ≈8.333\approx 8.333 m/s, still with no direction, while the velocity in (vi) is 20 m/s with a direction. The count: 4 scalars ((i), (ii), (iv), (v)) and 2 vectors ((iii), (vi)), 6 measures in all.

Common mistakes
  1. Calling 10 Newton a scalar because no direction is printed. The quantity is a force, and a force always acts in a direction; the book classes it as a vector.
  2. Mixing up speed and velocity. 30 km/hr with no direction is a speed (scalar); 20 m/s towards north is a velocity (vector).
  3. Reading the "3" in cm3^3 as a direction. It is the power of the unit; volume and density are scalars.
What to take away
  1. Classify the quantity behind the measure: time, volume, speed, density are scalars; force and velocity are vectors.
  2. A vector needs a direction to be complete, even if a measure prints only its size.
  3. Speed is a scalar; velocity, which is speed with a direction, is a vector.

Every fact used, and where it comes from

FactUsed inSource
A scalar involves only one value, its magnitude; lists of scalar quantities (time, speed, volume, density, …) and vector quantities (velocity, force, …)Parts (i) to (vi)Section 10.1
A quantity that has magnitude as well as direction is a vectorPart (iii); Part (vi), Steps 1, 2Definition 1, Section 10.2
A directed line segment represents a vectorPart (vi), Step 1 (figure)Fig 10.1(iii), Section 10.2 (Lesson: a vector is a length with a direction)
The book's classification of these six measuresResultExample 2, Section 10.3 (Lesson: Examples 1–3)
Vectors add by the triangle lawPart (vi), Step 2Section 10.4 (Lesson: adding arrows, head to tail)
Found an error, or want more?

Tell us if something on this page is wrong or unclear, or if you would like a fuller explanation, another method or a new feature. Send feedback on this problem →