A set of measured values of a quantity are recorded as having the values seven units, four units, negative two units, and six units. Which of the following types of quantities could these measurements represent? A) A vector quantity, B) A scalar quantity, C) Either a vector quantity or a scalar quantity, or D) Neither a vector quantity nor a scalar quantity.
Let’s start by recalling what scalar quantities and vector quantities are. A scalar quantity is any quantity that just has a magnitude or a size. Whereas a vector quantity has a magnitude and a direction in space. First, let’s think about whether these values — seven, four, negative two, and six — could possibly represent a vector quantity.
Now, you may have noticed that none of these measurements actually includes a direction. But we know that vector quantities have both magnitude and direction. However, this doesn’t necessarily mean that our measurements don’t represent a vector quantity. It might just be that there’s some extra information about the direction that we haven’t been told. In fact, these values do make sense as vector quantities.
For example, let’s imagine that all of these values are measurements of force, which is a commonly used vector quantity. We generally measure force in newtons, which would mean that these values become seven newtons, four newtons, negative two newtons, and six newtons. Now, while seven newtons, four newtons, and six newtons definitely make sense, negative two newtons definitely stands out as the only negative value in this list. But, in fact, this isn’t a problem. It’s actually common to use negative values of vector quantities to represent the fact that they act in the opposite direction to positive values.
For example, let’s consider the vertical forces acting on a rocket as it takes off. Let’s say that the rocket experiences a force of 10000 newtons upwards due to its engines and a force of 5000 newtons acting downwards due to gravity. In this case, if we wanted to do any arithmetic with these vectors, such as adding them together to find out the resulting force acting on the rocket. We would need to make sure that one of these values was negative, to reflect the fact that it acts in the opposite direction. In fact, whenever we use vectors, we need to define which direction is our positive direction and which direction is our negative direction.
In this case, if we said that our positive direction is up, which means that our negative direction is downwards, then we could say that the engines exert a force of positive 10000 newtons on the rocket and gravity exerts a force of negative 5000 newtons on the rocket. If we wanted to, we could just as easily define our positive direction as downwards, meaning our negative direction is upward. And then, we’d say that the engines are exerting a force of negative 10000 newtons on the rocket because the force is acting in the negative direction and gravity is exerting a force of positive 5000 newtons on the rocket. So we can actually use negative numbers to express some directional information about a vector quantity. And, in fact, all vector quantities can take either positive or negative values, which means that this list of values could be measurements of a vector quantity.
Now that we know this, we can rule out option D, neither a vector quantity nor a scalar quantity. And we can also rule out option B, which is just a scalar quantity. So now, we need to figure out whether these values could only represent a vector quantity or if they could represent a vector quantity or a scalar quantity. Now, we need to be a bit careful here because there are loads of scalar quantities for which negative values just don’t make sense. For example, mass, which is measured in kilograms. Now, we know that objects can and do have masses like seven kilograms or four kilograms or six kilograms. But there’s nothing known in physics that has a negative mass. So we know for sure that these values are not measurements of mass.
But there are actually some scalar quantities which can take negative values. For example, electrical charge, which is measured in coulombs represented by a capital C. Now, we know that some objects have a positive charge, and some objects have a negative charge. For example, the nucleus of an atom has a positive charge, whereas an electron has a negative charge. Which means it is possible to have measurements of charge equal to seven coulombs, four coulombs, negative two coulombs, or six coulombs.
But because charge doesn’t have a direction in space, it’s still very much a scalar quantity. Which means that this list of values could be measurements of a scalar quantity as well. So we know A is incorrect and the correct answer is C. The values seven units, four units, negative two units, and six units could represent either a vector quantity or a scalar quantity.