العلوم
Scalars and Vectors: The Difference Is Direction
A scalar has size only; a vector has size and direction. Ten quantities sorted into two columns, with one journey that has four different correct answers.
Scalar and vector quantities
A scalar is a quantity with magnitude only; a vector is a quantity that has both a magnitude and a direction, such as force or velocity.
- أين يقابله الطلاب
- Grade 8 science, and again in the first lesson of every GCSE and IGCSE physics course, because the whole forces and motion unit is built on it.
الإجابة باختصار
A scalar is a quantity with size only; a vector has both a size and a direction. Mass, time, speed, distance, energy and temperature are scalars. Displacement, velocity, force, acceleration, momentum and weight are vectors. The test is whether the quantity would be incomplete if the direction were left out.
مثال
One lap of a 400 m track in 50 s: distance 400 m, displacement 0 m; average speed 8 m/s, average velocity 0 m/s.
The runner has certainly moved, and 400 m of track has certainly passed under their feet — that is the distance, and it is a scalar, so it just accumulates. Displacement asks a different question: how far, and in what direction, are you from where you started? The answer is nowhere, so the displacement is zero, and the average velocity, which is displacement divided by time, is zero as well. Neither answer is a trick. They are answers to two different questions.
Twelve quantities, sorted
Learning the definition is easy and applying it is not, so it is worth sorting a list once, carefully, and keeping it. The rule to apply each time: would the quantity still be fully specified if nobody told you which way?
One pair on this list catches almost everyone. Mass is a scalar — 60 kg, and there is nothing more to say. Weight is a vector, because it is a force, and it points downwards, towards the centre of the Earth.
- Scalars: distance, speed, mass, time, energy, temperature
- Vectors: displacement, velocity, force, acceleration, momentum, weight
- The pairs to keep straight: distance and displacement, speed and velocity, mass and weight
One journey, four correct answers
Walk 3 km east, then 4 km north, taking one hour in total. The distance travelled is 7 km, because distance simply adds up whichever way you turn. The displacement is the straight line from start to finish, which by Pythagoras' theorem is 5 km, on a bearing of about 037°.
Divide each by the hour and the two speeds follow: an average speed of 7 km/h, and an average velocity of 5 km/h on that bearing. Four numbers, one walk, and no contradiction between them — which is why an exam question that says velocity and gets answered with a speed cannot be given the mark.
Vectors can cancel; scalars cannot
Two forces of 5 N can add to 10 N, or to 0 N, or to anything in between, depending on the directions they act in. Two energies of 5 J always make 10 J. That difference is the whole reason resultant force calculations exist as a topic, and the reason a tug of war can be perfectly balanced while both teams pull as hard as they can.
It is also why direction has to be tracked rather than assumed. Once a direction is chosen as positive, everything the other way carries a minus sign, and the arithmetic then handles cancellation on its own. A single sign, chosen once at the start of a question and then used consistently, does more work than any amount of reasoning about which way things are pointing.
أسئلة شائعة
Is weight a scalar or a vector?
A vector. Weight is the force of gravity acting on a mass, measured in newtons, and it acts downwards. Mass is the scalar: it is a measure of how much matter there is, measured in kilograms, and it is the same on the Moon as on Earth while the weight is not.
Is speed the same as velocity?
No. Speed is how fast, velocity is how fast and in which direction. A car doing 30 m/s round a bend has a constant speed and a changing velocity. In everyday conversation the words are interchangeable; in a physics answer they are not, and swapping one for the other loses marks.
Can a vector be negative?
In one dimension the minus sign is how direction is written. If right is taken as positive, then −5 m/s means 5 m/s to the left; the size is still 5. In two or three dimensions direction is given by an angle or a bearing instead, and the size on its own is never negative.
Is temperature a vector, since it goes up and down?
No. Up and down here describe an increase or a decrease in a number, not a direction in space. A vector needs a direction you could point at. Temperature has a size and a sign, which is not the same thing — −5 °C is a temperature, not a temperature pointing somewhere.
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