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Vector vs scalar: what's the difference?

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University Course Reader · STEM

Velocity is a vector quantity, so two cars at the same speed in opposite directions have different velocities.

The reader highlighted one word in a textbook passage. Clicked broke down the physics term “vector” into plain English:

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Overview

A scalar is a quantity that a size alone describes completely, like temperature, mass or time. A vector is a quantity that needs a size and a direction, like force, velocity or displacement. The test is to ask which way. "Which way is 20 degrees?" is a meaningless question, so temperature is a scalar. "Which way is the force?" is meaningful, so force is a vector.
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Overview

A scalar is a quantity that is just a size: 20 degrees, 70 kilograms, ten minutes. A vector is a size that comes with a direction attached: 60 kilometres an hour north, 10 newtons to the left. Speed is a scalar and velocity is a vector, which is why a runner lapping a track at a steady pace has a constant speed and a velocity that changes at every bend. Ask "which way?": if there is an answer, it is a vector; if the question is nonsense, it is a scalar. 😎

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Detail

A scalar is a quantity that a size alone describes completely: 20 degrees, 70 kilograms, four seconds. A vector is a quantity that needs a size and a direction before it is described at all: a force of 10 newtons to the left, a velocity of 60 kilometres an hour north. Speed and velocity are the pair most people mix up. Speed is a scalar, the number on the speedometer. Velocity is that number plus a heading. A car circling a roundabout at a steady 30 kilometres an hour has constant speed and a velocity that changes the whole way round, because its direction never stops changing. The difference matters because vectors combine by direction. Two 10 newton forces pushing the same way add to 20 newtons; the same two forces pushing opposite ways add to nothing, and the object sits still. Two 10 kilogram bags are 20 kilograms whichever way you hold them, because mass is a scalar and scalars just add. The direction has to be a direction in space. "The temperature is falling" describes a scalar getting smaller, and "five kilograms of flour" points nowhere, so neither is a vector.
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Detail

A scalar is a quantity that is only a size, and a vector is one that has a size and a direction. Mass, temperature and time are scalars; force, velocity and displacement are vectors. The reason anyone bothers with the distinction is that the two kinds combine differently. Scalars add like shopping: 5 kilograms of potatoes plus 5 kilograms of onions is 10 kilograms, and that is all there is to it. Vectors add by direction, so the same two numbers can give you anything from their sum to nothing. Two people hauling a sledge from the front with 200 newtons each give it 400. Have one of them pull the other way and the same 400 newtons of effort adds up to zero, and the sledge stays put. Speed and velocity are the same distinction on a running track. Speed is the number on the runner's watch, a scalar. Velocity is that number with a heading, and heading counts: a runner lapping the track at a steady pace keeps her speed and changes her velocity at every bend, because she keeps turning. Not every number that sounds directional is a vector. "The temperature is dropping" is a scalar getting smaller, and there is no such thing as 20 degrees to the north. 😎

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Analogy

A scalar is telling a lost stranger "it is two kilometres". A vector is telling them "it is two kilometres north". The number is the same in both. The distance tells them how far, and only the direction tells them where, so only the second answer gets them there. Temperature, mass and time are the first kind of answer: complete on their own. Force, velocity and displacement are the second kind: unfinished until you say which way. Ask "which way?" of any quantity. If the question has an answer, the quantity is a vector; if the question makes no sense, it is a scalar.
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Analogy

A scalar is a size, a vector is a size with a direction, and two dogs on leads will show you the difference. Each dog pulls with the same strength. Both pull towards the park and you get dragged; one pulls towards the park and one towards home and you stand still, leaning. The pulls were the same size both times, and their direction decided whether they added up or wiped out, which is what makes the force of a pull a vector. Their dinner is a scalar: two bowls of 200 grams is 400 grams of dog food, and it stays 400 grams whether they pulled you to the park or held you on the pavement. Forces point, so forces can cancel; grams just add up. 😎
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Formal definition — The same term, explained the usual way

A scalar is a physical quantity that is completely specified by its magnitude, expressed as a real number with appropriate units; examples include mass, temperature, time, energy, distance and speed. A vector is a physical quantity that requires both a magnitude and a direction for its complete specification; examples include displacement, velocity, acceleration, force and momentum. Scalars combine by ordinary arithmetic, whereas vectors combine according to the rules of vector addition, in which the directions of the components determine the magnitude and direction of the resultant. Speed is the scalar magnitude of the velocity vector, and distance is the scalar magnitude of displacement only when motion is in a straight line.

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