Scalars and vectors

Two people push a car that has broken down, each with a force of three hundred newtons. Push from the back together, and the car rolls forward. Put one of them at the front, pushing back, and the car does not move at all. The two numbers never changed, to the newton. Something other than the number decided what the car did.

This is video two of six in Units and describing motion, for combined science and triple physics, Foundation and Higher alike. It does not lean on the unit video, so you can start right here.

A scalar is a quantity with a size only. A vector is a quantity with a size and a direction, and it is not complete without both. The size has a proper name, magnitude. Magnitude means how big it is, ignoring which direction it points, and it always comes with its unit. Picture two readings side by side. A balance reads two kilograms. A newton meter, the spring gauge that measures force, reads twenty newtons. Both readings are a number with a unit, and the two displays look alike. But the newton meter is being pulled along its spring, in one direction, and its display cannot show you that. Force is a vector. Twenty newtons is only half of it, and the other half is the direction, such as twenty newtons downwards. Mass has no direction at all, so mass is a scalar. That gives you one test for any quantity. Does it need a direction before it means anything? If it does, it is a vector. If it does not, it is a scalar. Now try temperature. A room is at twenty degrees Celsius: scalar or vector? Scalar. Twenty degrees is complete as it stands, because a temperature cannot point anywhere.

The scalars a paper expects you to know are mass, temperature, time, energy, speed and distance. Not one of them needs a direction. The vectors are force, weight, velocity, displacement, acceleration and momentum. Weight is a force pulling down, which is why it sits beside force. Distance and displacement get a video of their own later in this chapter, and so does velocity. For now, each one only needs its column. Which one is a vector: a battery's stored energy, a race time, or friction on a sliding box? The friction. Friction is a force, and it always acts against the sliding, so it has a direction. The energy and the time have none. One examiner's report on a Foundation paper puts it like this. The majority of candidates appeared to struggle to apply their knowledge of scalar quantities and vector quantities to how the information was presented in the table. The same report goes on. Candidates who wrote down the definition of each quantity next to the question were more likely to choose the correct option. Knowing the two words was not the problem, using them on a real list was. The fix is that habit: write both definitions beside the question, then run the direction test on each item.

A vector can be drawn as an arrow. The length of the arrow shows the magnitude, drawn to a stated scale, and the arrowhead shows the direction. Use a scale of one centimetre for one newton. A six newton force pointing right is an arrow six centimetres long, with its head on the right. A three newton force pointing left is an arrow three centimetres long, with its head on the left. It is half the length, because it is half the force. Now read an arrow back the other way, from its length to its force. At one centimetre to two newtons, a four centimetre arrow points down. What force? Eight newtons, downwards. Four centimetres at two newtons per centimetre makes eight, and the arrowhead gives downwards. A direction can be given in three ways. A compass point, like north. A bearing, which is an angle measured clockwise from north. Or a sign, once you have said which direction counts as positive. Two people push a box, twenty newtons to the right and twelve newtons to the left. Take right as positive. The first push is plus twenty newtons, and the second is minus twelve newtons. The minus sign does not make twelve smaller. It is a direction, and it says this push is to the left. How two pushes combine is a job for the forces chapter, so here they stay apart.

A tow truck pulls a broken-down van along the road with a rope. Someone describes the pull as four kilonewtons, and stops there. What is missing from the answer four kilonewtons? The direction. Four kilonewtons is only the magnitude. A complete answer is four thousand newtons, forwards. Here is a line from a report on a Higher paper, about an answer that needed a vector. Only three per cent of students scored two marks for adding that the direction does not change. In plain words, nearly everyone left the direction out, even though it had stayed the same. The fix is to give every vector sentence its second half, out loud, even when the direction has not changed. Your handle for this video comes in two parts: the size, and the direction.

Three to check yourself on before you go, and none of them has appeared in this video. A jet engine's thrust is a push. Is thrust a scalar or a vector, and why? A vector. Thrust is a force, so it needs a direction, forwards, before it means anything. Try this one. A lift cable pulls with six thousand newtons. What is the complete answer? Six thousand newtons, upwards. A cable can only pull the lift up, and upwards is the second half of the sentence. Here's another. What does a vector arrow's length show, and what does its head show? The length shows the magnitude, to the stated scale. The head shows the direction. And the broken-down car: three hundred newtons each time, and the only thing that changed was a direction. That is why force is a vector, and why every vector answer takes the size and its direction.

Once you could teach this one, thumb it up, so you know it never needs another watch. If it has not landed, save it and write the two definitions out once; that makes them stick.

Next in the chapter: Distance and displacement, where one journey gives two different answers.

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