What is a vector?

From PhysicsEd

In physics, all physical quantities can be placed into one of two categories: vector quantity or scalar quantity. A vector is a physical quantity that has both direction and magnitude. For example, velocity, acceleration and displacement. They all are specified by the magnitude, or size, and the direction of that magnitude. A scalar has no direction and is only specified by the magnitude (e.g temperature, mass or volume). Vectors are often written with a small arrow over the letter.


The arrow over the symbol means it has direction.

We can break vectors up into components. A vector can be completely described by its components. We can take any vector on an xy-plane and break it into the x-component and y-component. This is simply saying how much of the arrow lies in the x-direction and how much in the y-direction.

\vec{V} = V_{x} + V_{y}

V_{x} is the x-component of the vector. V_{y} is the y-component of the vector.

Numbers with direction become vectors which can describe forces, velocities, gravitational fields, magnetic fields, electric currents, a position in space and time, and even string theory of the universe in many dimensions.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \vec{V} = a_{x} \vec{V} = 3

Adding vectors is easy, too, because you can add their components:

V⃗ =3x^+4x^

V⃗ =7x^

An exercise to think about

Suppose a policeman chases a burglar in an airport in the wrong direction on a moving sidewalk. The sidewalk is moving 3 km/hr to the right, and both the policeman and the burglar are walking 2 km/hr to the left; how fast is the burglar moving with respect to the airport? Everything is along a line that we call x^ increasing to the right:

V⃗ =ax^+bx^

V⃗ =3x^+−2x^

Credit: The New Yorker

V⃗ =?x^

Answer: V⃗ =1x^

The burglar and the policeman are both moving backwards to the right at 1 km/hr with respect to the airport although they are walking at 2 km/hr to the left on the walkway. Direction is important, even on a straight line. What should the policeman do if he wants to catch the burglar?

What about a vector in 2 dimensions? It may be graphed using its x and y components like this, but how long is it?

V⃗ =ax^+by^

V⃗ =4x^+3y^

Since the component vectors make a right triangle with the vector, the ancient (over 2,500 years old) Pythagorean Theorem may be used.

So . . .

Exercise to help you see how this works:

Suppose x = 3 and y = 4. What is the length of the vector V ?

Answer: Fortunately we know the square root of 25 is 5.

Since the vector forms a right triangle with its component vectors, and because you can now find the lengths of x, y, and V, you can also know the sine, cosine, and tangent of the angle.

From these you can find the arcsine, arccosine, and arctangent which will give you the angle. ("Arctangent" has the meaning, "What angle has a tangent with this value?" For this you need a calculator.

Suppose x = 4, y = 3, and h = 5

Using the sine, you divide 3 by 5, and the calculator will give you the sine in decimal form.

To find the arcsine of that number, press inverse (inv) then sine (sin)

Voila! you have the angle, ~36.8698976458 degrees

Exercise to see if you can do it another way:

Find the angle, using arctangent, where y = 3 and x = 4.

Answer: 3 divided by 4 = 0.75

press "inv" or the "-1" button, then "tan"

Voila! the angle is roughly 36.8698976458 degrees

There is a very useful web calculator that you might bookmark because it also includes math help, and a tool to plot a calculation.

Lorentz transformations in Special Relativity have 4 dimensions, the fourth being Time. We cannot "see" 4 dimensions, but we can do the math and also create drawings to help us with a mental picture, like the 4D cube below:

Credit: Wikipedia

And String theory has 26 spacetime dimensions for the bosonic string and 10 for the superstring, to make a consistent quantum theory !

Credit: Wikipedia

Ordered set notation

A vector can be specified using its components from each dimension, enclosed in either parentheses or angle brackets.

V = (x, y, z, etc.)


where n is the number of components of V.

This makes it easy to keep track of the parts that describe a vector. We can do the math on each part separately. We'll see how this works later in the course when we describe motion through space and the effects of gravity.

Matrix notation

In case you have seen this in a math class before, it may help to know that a vector in can also be specified as a row or column matrix containing the ordered set of components. A vector specified as a row matrix is known as a row vector; one specified as a column matrix is known as a column vector. Again, an n-dimensional vector can be specified in either of the following forms using something like this. You can use this notation if you prefer and sometimes operating on vectors this way is easier that writing out their components.

A vector written as a column matrix


Credit: Wikipedia