What is a Vector?

What is a vector? A vector is fundamentally a simple concept used broadly in modern science, (from physics, to virology, mathematics and quantum mechanics). When we first start learning new things, the unknowns make something more complicated than it actually is — or will be after a couple of months of study.

This happened to me with vectors.

Simply put, a vector is a quantity that has both magnitude and direction. It refers to any type of physical quantity that is in possession of two fundamental characteristics — essentially a size and direction.

For example, in physics, velocity is different than speed. Speed is what your odometer tells you — that you are cruising on the highway in your car at 120 km/h, or in your race car on a track at 200 mph.

However, those values do not tell you anything about where you are going. Exactly in what direction you are travelling at the same time, precisely according to GPS coordinates over intervals of time and a heading according to precise compass directions. You might have a general reading in your vehicle like North (N) or Northwest (NW) — but not down to a precision of $271 \deg$ northeast.

Velocity on the other hand — velocity according to physics — tells both the speed and direction of a moving body. That’s what makes velocity a vector and not a scalar.

If an airplane is flying through the air we can denote its velocity vector in some form of diagram with an arrow (like the one above) point from its nose exactly in the direction it is travelling with the length of the arrow precisely denoting the speed of the plane, such that if the airplane were to travel half its current speed, then the arrow would be half the length of the original. Whereas if the plane were to double its speed then the arrows length would also double.

Vectors are quantities depicted visually with arrows. They are physical quantities that possess both magnitude and direction. The direction of the arrow denotes the direction in which the physical quantity is acting, while the length of the arrow denotes the magnitude of the physical quantity acting.

Vectors in the Real World

Many of the fundamental physical quantities (found in the SI Units) are vectors; they possess both magnitude and a direction. This is what distinguishes them from the other main type of physical quantity called the scalar, which only possesses a magnitude (or size).

Quantities like mass, temperature, volume, length, speed, or time are all scalars, or scalar quantities, because you can refer to them with a single number a unit. They only possess magnitude. Some of the most elementary physical properties that we study are vectors. This includes velocity, displacement, acceleration, force, momentum, electric force, magnetic force, and the orientation of both electric and magnetic fields.

Vectors are fundamental to physics. Scalar products of vectors determine and define other scalar quantites, such as energy. While vector products of vectors define other vector physical quantities such as torque or angular momentum.

Vector Notation

At this stage of our learning vectors are Euclidean quantities which have geometric representation. In diagrams we can refer to and define a vector as an arrow, where both the size and direction of the arrow specifically delineates a quantity.

They are represented with an arrow and are measured from tail-to-head or from tail-to-tip.

We can represent them graphically as an arrow in 1D, 2D, or 3D. They can be added, subtracted, or multiplied.

Vector operations also have numerous other generalizations in many branches of physics.

In mathematics and physics texts it is common to denote a vector quantity using boldfaced letters. For example, we may right velocity as a bold “v”, $\mathbf{v}$, though it is most common to distinguish vectors with the arrow over the symbol $\vec{v}$. Many texts will do both as in $\mathbf{\vec{v}}$ (which is the method that I use here).

Further Reading

• Vectors
  • What is a Vector?
  • Adding and Subtracting Vectors
  • Multiplying Vectors

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