Vectors & Scalars: What Are They & Why They Matter?

In everyday life, most people use the terms ​speed​ and ​velocity​ interchangeably, but to physicists, they're examples of two very different types of quantity.

Mechanics problems deal with the motion of objects, and while you can just describe motion in terms of speed, the specific direction that something is going is often critically important.

Similarly, the forces applied to objects can come from many different directions – think about the opposing pulls in a tug of war, for instance – so physicists describing situations like this need to use quantities that describe both the "size" of things like forces and the direction in which they act. These quantities are called ​vectors​.

The key difference between vectors and scalars is that a vector's magnitude doesn't entirely describe it; there also needs to be a stated direction.

The direction of a vector can be stated in numerous ways, whether through positive or negative signs in front of it, expressing it in the form of components (scalar values next to the appropriate ​​i​​, ​**​j​**​ and ​​k​​ "unit vector," which correspond to the Cartesian coordinates of ​x​, ​y​ and ​z​, respectively), adding an angle with respect to a stated direction (e.g., "60 degrees from the ​x​-axis") or simply adding some words to describe the direction (e.g., "northwest").

By contrast, a scalar is just the vector's magnitude without any additional notation or information provided – for example, speed is a scalar equivalent of the velocity vector. From a mathematical perspective, it's the absolute value of the vector.

However, many quantities, such as energy, pressure, length, mass, power and temperature are examples of scalars that aren't just the magnitude of a corresponding vector. You don't need to know the "direction" of mass, for example, to have a complete picture of it as a physical property.

There are a few counterintuitive facts that you can understand when you know the difference between a scalar and a vector, such as the idea that something could have a constant speed but a continuously changing velocity. Imagine a car driving at a constant speed of 10 km/h but in a circle. Because the direction of a vector is part of its definition, the car's velocity vector is always changing in this example, despite the fact that the magnitude of the vector (i.e., its speed) is constant.

There are many examples of vectors in physics, but some of the most well-known examples are force, momentum, acceleration and velocity, all of which feature strongly in classical physics. A velocity vector could be displayed as 25 m/s to the east, −8 km/h in the ​y​-direction, ​**​v​​ = 5 m/s ​i​ + 10 m/s ​j**​, or 10 m/s in a direction 50 degrees from the ​x​-axis.

Momentum vectors are another example you can use to see how the magnitude and direction of the vector are displayed in physics. These work just like the velocity vector examples, with 50 kg m/s to the west, −12 km/h in the ​z​ direction, ​**​p​​ = 12 kg m/s ​i​ – 10 kg m/s ​j​ – 15 kg m/s ​k**​ and 100 kg m/s 30 degrees from the ​x​-axis being examples of how they could be displayed. The same basic points go for the display of acceleration vectors, with the only difference being the unit of m/s² and the commonly-used symbol for the vector, ​​a​​.

Force is the final one of these examples of vector expressions, and while there are many similarities, using cylindrical coordinates (​r​, ​θ​, ​z​) instead of Cartesian coordinates can help to show other ways they may be displayed. For example, you might write a force as ​**​F​​ = 10 N ​r​ + 35 N ​𝛉**​, for a force with components in the radial direction and the azimuthal direction, or describe the force of gravity on a 1-kg object on Earth as 10 N in the –​r​ direction (i.e., towards the center of the planet).

In diagrams, vectors are displayed using arrows, with the magnitude of the vector represented by the length of the arrow and its direction represented by the direction in which the arrow points. For example, a larger arrow shows that a force is larger (i.e., more newtons or a bigger magnitude) than another force.

For a vector that shows motion, such as the momentum or the velocity vector, the ​zero vector​ (i.e., a vector representing no velocity or momentum) is displayed using a single dot.

It's worth noting that because the length of the arrow represents the magnitude of the vector and its orientation represents the direction of the vector. It's useful to try to be reasonably accurate when making a vector diagram. It doesn't have to be perfect, but if the vector ​​a​​ is twice as big as the vector ​​b​​, the arrow should be roughly twice as long.

Vector addition and vector subtraction are a bit more complicated than adding and subtracting scalars, but you can pick up the concepts easily. There are two main approaches you can use, and each has potential uses depending on the specific problem you're tackling.

The first, and the easiest to use when you've been given two vectors in component form, is to simply add matching components in the same way you'd add ordinary scalars. For example, if you needed to add the two forces ​**​F​**​₁ = 5 N ​i​ + 10 N ​j​ and ​**​F​**​₂ = 6 N ​i​ + 15 N ​j​ + 10 N ​k​, you would add the ​i​ components, then the ​j​ components and finally the ​k​ components as follows:

\(\begin{aligned}
\bm{F}_1 + \bm{F}_2 &= (5 \;\text{N} \;\bold{i} + 10 \;\text{N}\;\bold{j}) + (6 \;\text{N} \;\bold{i} + 15 \;\text{N}\;\bold{j} + 10 \;\text{N}\;\bold{k})\)
\(&= (5 \;\text{N} + 6 \;\text{N}) \bold{i} + (10 \;\text{N} + 15 \;\text{N}) \bold{j} + (0 \;\text{N} + 10 \;\text{N}) \bold{k}\)
\(&= 11 \;\text{N} \;\bold{i} + 25 \;\text{N} \;\bold{j} + 10 \;\text{N} \;\bold{k}
\end{aligned}\)

Vector subtraction works in exactly the same way, except you subtract the quantities rather than add them. Vector addition is also commutative, like ordinary addition with real numbers, so ​**​a ​​+ ​​b​​ = ​​b​​ + ​​a​**​.

You can also perform vector addition using arrow diagrams by laying the vector arrows head to tail and then drawing a new vector arrow for the sum of the vectors connecting the tail of the first arrow with the head of the second.

If you have a simple vector addition with one in the ​x​-direction and another in the ​y​-direction, the diagram forms a right-angled triangle. You can complete the vector addition and determine the resulting vector's magnitude and direction by "solving" the triangle using trigonometry and Pythagoras' theorem.

Multiplying vectors is a bit more complicated than scalar multiplication for real numbers, but the two main forms of multiplication are the dot product and the cross product. The dot product is called the scalar product and is defined as:

\(\bm{u} \;∙ \;\bm{v} = u_1v_1 + u_2v_2 + u_3v_3\)

or

\(\bm{u} \;∙ \;\bm{v} = \lvert\bm{u}\rvert\lvert\bm{v}\rvert \text{cos}(θ)\)

where ​θ​ is the angle between the two vectors, and the subscripts 1, 2 and 3 represent the first, second and third component of the vector. The result of the dot product is a scalar.

The cross product is defined as:

\(\bm{a} \; \bold{×} \;\bm{b} =(a_2b_3 − a_3b_2, a_3b_1 − a_1b_3,a_1b_2 − a_2b_1)\)

with the commas separating components of the result in different directions.