In this explainer, we will learn how to distinguish between scalar quantities with magnitudes and vector quantities with both directions and magnitudes.
Let us start by looking at some examples of physical quantities to distinguish whether each one has a directional property.
Example 1: Identifying Scalar and Vector Quantities
Which of the following is a vector quantity?
- Potential difference
Remember that a vector has both magnitude and direction, unlike a scalar, which has only magnitude. Let us think about some possible values that these quantities could have, expressed in SI units. For example, we could have 200 J of energy, 12 V of potential difference, C of charge, a force of 50 N upward, or 900 Pa of pressure.
These quantities all have magnitudes, but in order to be a vector, a quantity must also include a direction. Let us determine whether it makes sense to assign a direction to each of these quantities. Energy measures an object’s ability to do work by virtue of its motion, position, or internal properties. For example, if we know the ball on the hill in the diagram below has 200 J of potential energy, we know the magnitude of the energy, but we cannot meaningfully assign it a direction. The ball has the ability to roll down the hill in any direction, but all paths down the hill are indistinguishable in terms of energy because the quantity is not associated with any particular direction.
Potential difference and charge are quantities associated with the electric force, which can point in any direction. For example, a positive point charge creates an electric field that points radially outward, as illustrated by the black arrows in the figure below. If any positive test charge was placed nearby, the positive charge would exert a repulsive force on it, as represented by the purple arrows, resulting in the negative charge accelerating through a potential difference. However, the direction of the repulsive force depends on the positions of the positive and negative charges, and it is neither the charge nor the potential difference that actually points in a direction. Thus, it is not meaningful to associate either quantity with a direction.
Pressure is a force spread over an area. Imagine pushing down on the lid of a container filled with putty, with holes on all sides of the container, as shown below. Notice how pushing down on the lid applies pressure to all of the putty in the container, as represented by the yellow arrows, and the putty flows out through the holes in all sides of the container. Pressure is applied in all directions at once, and so we cannot associate it with any one specific direction and instead represent pressure as a scalar quantity.
The only quantity listed above that has a direction associated with it is force, as it is specified as exerted upward. Imagine if a ball is thrown with a force of 50 N upward, and another ball is thrown with a force of 50 N to the right, as illustrated below. Even though they have the same magnitudes, the forces can be distinguished from one another in a way that scalars cannot. Here, a difference in direction is a meaningful part of measurement. In order for a force to be fully described, both magnitude and direction must be specified.
Therefore, force is a vector quantity.
Now that we have seen examples of both types of quantities, let us look at their formal definitions.
A scalar quantity expresses only magnitude, or how much of a quantity there is. A scalar can have units to help describe its magnitude, but not a direction.
A vector quantity expresses both magnitude and direction. A vector can have units to describe its magnitude, and the quantity must be associated with a specified direction.
Although these two terms appear similar, their key difference is that a vector is assigned a specific direction, and a scalar is not.
Remember that units can be assigned to both types of quantities. For instance, if we know that a dog ran 20 m, how far it ran is expressed as a scalar. But, if our friend said she saw the dog run 20 m toward the park, then we have a vector quantity that we can use to describe the dog’s displacement, which tells how far it ran in a certain direction. Both of these examples of a scalar and vector have the same unit, metres, but produce different results, since specifying a direction tells us something more about the dog’s motion.
Example 2: Identifying Scalar and Vector Quantities
A set of measured values of a quantity are recorded as having the values 7 units, 4 units, negative 2 units, and 6 units. Which of the following types of quantities could these measurements represent?
- A vector quantity only
- A scalar quantity only
- Either a scalar quantity or a vector quantity
- Neither a scalar quantity nor a vector quantity
To begin, let us recall that scalars are quantities which only have magnitude and vectors are quantities which have magnitude and direction and notice that none of the values listed above include a written direction. However, this does not mean that these magnitudes may never be associated with a direction, so let us imagine whether it would make sense for vector quantities to take on the values , and 6.
Force is a vector quantity commonly used throughout physics, so imagine we have the values expressed in the SI unit of force, newtons, as 7 N, 4 N, N, and 6 N. The negative value may raise some suspicion, but it should be noted that negative vector quantities often represent values that point in the direction opposite to the positive forces. For instance, imagine pushing a box across a floor with a force of 100 N to the right, while friction on the box causes a resisting force of 25 N to the left. If we find the net force acting on the box by choosing to represent the rightward direction as positive and the left as negative, we can calculate the net force as . Thus, we know that the box has a net force of 75 N acting on it toward the right.
We were able to meaningfully assign a negative value to a vector, and so it is reasonable to conclude that the set of values , and 6 might represent a vector quantity.
Now we must decide whether our set of values could potentially represent a scalar quantity, so let us consider what it would mean for a scalar to have a negative value. While some units of measurement, such as time, cannot meaningfully assume a negative value, there are some scalar quantities which can reasonably have negative values. For instance, consider electric charge: protons have a charge value of C, and electrons have charge C. Notice that the values have the same magnitudes, but because one is positive and one is negative, it is important that they are distinguished. Therefore, charge is a scalar quantity that can meaningfully have a negative value.
Therefore, it is possible that the set of measured values , and 6 could represent either a scalar or vector quantity, which corresponds to answer choice C.
As demonstrated above, we encounter scalar and vector values in real life, throughout our study of physics, so it is useful to consider how these types of quantities interact mathematically.
A vector can be represented graphically, using an arrow, such that the length of the arrow represents the vector’s magnitude, and the arrow points in the direction of the vector. This representation can help us better understand vector properties and how they interact with other quantities. The following example demonstrates the “tip-to-tail” method of adding vectors, which results in a new vector.
Example 3: Identifying the Correct Graphical Representation of the Addition of Vectors
The two vectors shown can be added to produce a resultant vector. Which of the following diagrams correctly shows a comparison of both of these vectors to the resultant of the vectors?
Note that vector lengths are not to scale.
When adding vectors graphically, we use the “tip-to-tail” method.
Vectors are added by the commutative property, which means they will always result in the same sum, no matter the order in which they are added. So, beginning with either vector, locate its “tip” (the end with the arrowhead). Then locate the “tail” (the end without the arrowhead) of the other vector. We will use the tip of the blue vector and the tail of the green vector, as shown below.
Next, translate, or slide without rotating, the second vector so that its tail is located at the same point as the tip of the first vector. This step is illustrated below.
Now that the two are arranged “tip-to-tail,” we can draw a new vector to represent their sum, beginning at the tail of the first vector and ending at the tip of the second vector. This sum is represented by the yellow vector below.
Further, notice that because of the commutative property, we could have added the vectors in a different order and achieved the same sum, as shown below.
Thus know that answer choice B correctly shows how to add the two vectors.
When adding and subtracting scalars or vectors, the rule is simple: we must be sure to only add or subtract like quantities. It is not possible to meaningfully add or subtract two quantities with different units. For example, try adding 25 joules and 10 hertz into one understandable term and it will become clear that adding terms with mismatched units is not at all useful.
Further, it is not possible to add or subtract a scalar with a vector. Even if the two quantities use the same units, scalars and vectors are still fundamentally different from one another; if we do not have a directional component, or element, for one quantity, then we cannot justify adding or subtracting it with a quantity that does involve a direction, since we would not have all the information necessary to find a true and complete result.
Let us think of a situation in which we might try to add a scalar and vector together. Imagine we find directions to a hidden treasure which read, “Walk 60 steps, then walk 40 steps west.” We would know the exact location of the treasure if we could calculate its total displacement from the given instructions, which would take the form: 60 steps 40 steps west. But there is a problem, because we do not know the direction in which to walk the first 60 steps, so there is no clear interpretation of adding the two quantities together. We could guess, and perhaps try walking the first 60 steps north, south, east, or west, but it is also possible that the steps need to be taken in some in-between direction. This means there are many different paths that could be attempted, while only one path actually works. If only the instructions included an initial direction along with the magnitude of 60 steps, we could have known the treasure’s location immediately.
The key point to remember when adding or subtracting is that we must use like terms: scalar with scalar, or vector with vector, always making sure the involved units match.
It should be noted that it is not possible to divide by a vector, so we will not attempt to do so. But, perhaps surprisingly, it is possible to divide a vector by a scalar and to multiply scalars and vectors together. To do this, we multiply or divide the quantities’ magnitudes like normal and also multiply or divide their units, taking care to consider what this change of units means for the resulting quantity. The direction of the vector quantity still applies to the product or quotient, so multiplying or dividing scalars and vectors together results in a vector. Representing vectors graphically is helpful in understanding this concept, as the multiplication or division of a vector by a scalar just involves changing the length of an arrow, leaving it as a vector.
Imagine we know that a bicycle is traveling to the right at , and we want to find out its displacement after 2 s. Because it is a vector quantity, we can represent the velocity of the bicycle with an arrow, such as the blue arrow below, which points to the right and is 8 units long.
Time is a scalar quantity, so it cannot be represented by an arrow. However, we can still perform a mathematical operation with our vector and scalar. In order to find the displacement of the bicycle, we need to multiply velocity by time, so we multiply the magnitude of the vector by two, which just doubles the arrow’s length. This product is represented by the green arrow below, and the direction is still relevant, since displacement is a vector quantity. So, after two seconds, the bicycle has a displacement of 16 m to the right.
It is important to remember that multiplying or dividing a vector by a scalar results in a new vector with the same direction, but a different magnitude. Let us apply this concept to some examples.
Example 4: Determining Whether the Product of Two Quantities Is a Scalar or Vector
If a speed is multiplied by a time, is the resultant quantity a vector quantity or a scalar quantity?
We want to multiply a speed by a time; let us first think about the properties of these quantities. We must be careful not to confuse speed with velocity, its vector counterpart. Speed has no directional element, so it is a scalar quantity.
Time is also a scalar quantity, so we want to multiply two scalars together, which we are already familiar with: simply multiply their magnitudes and units. Since there is no direction associated with time or speed, their resulting product will be a scalar, too.
Example 5: Determining Whether the Product of Two Quantities is a Scalar or Vector
If a time is multiplied by a velocity, is the resultant quantity a vector quantity or a scalar quantity?
We want to multiply a time by a velocity, so to check whether this is valid, let us think about the two quantities and their properties. Time is a scalar quantity, since it can be fully described using only magnitude, and remember that velocity is a vector quantity:
Not to be confused with speed, its scalar counterpart, velocity is defined as displacement divided by time, and remember that displacement is a vector because it has both magnitude and direction.
We want to multiply a scalar by a vector, which can be done by multiplying their magnitudes and units and assigning the original direction of the vector to the product. Since we will end up with a direction associated with our product, the resulting quantity is a vector.
We can double check this—multiplying time and velocity gives us
The units of time cancel out and our quantity turns into displacement, which has both magnitude and direction, so the resulting quantity is a vector.
Note that, as mentioned in the example above, the very definition of velocity involves dividing a vector (displacement) by a scalar (time), resulting in a vector quantity, just like we saw before with the bicycle.
Let us finish by summarizing a few important concepts from this explainer.
- Scalar quantities have only magnitude, and vector quantities have both magnitude and direction.
- When adding and subtracting, we must use either vector and vector or scalar and scalar. In both cases, all involved units must match.
- We can add vectors graphically “tip-to-tail,” creating a new vector.
- We can determine whether calculated values using scalar and vector quantities have a scalar or vector value.
- When multiplying or dividing a vector and scalar together, multiply or divide their magnitudes and units, and the product will keep the vector’s directional element, resulting in a vector.