Mathematical methods

9. An introduction to scalars and vectors


The idea of scalars and vectors is, arguably, the most important one for anyone interested in physics to fully understand. It can be curious at first glance but, as time goes this simple idea will bloom and help solve some of the toughest problems one might encounter in physics.

Definitions

Scalars

A scalar is simply a physical quantity without a direction associated with it. For instance, the temperature outside may be 24°C, but not 24°C towards northeast, or 24°C at a 45° angle from the roof and so on. In other words, temperatures have values independent of any direction. The same goes for mass, length, density etc. These are all scalars.

In print scalars are generally represented in normal weights as opposed to vectors (see below). When handwritten they are variables like any other.

Vectors

A vector, on the other hand, compulsorily has a direction — it often makes no sense without a specific direction. For instance, one could say a car is travelling at 80 kmh$^{-1}$, but little about the car can be determined until we know what direction it is travelling in. Similarly, a person’s weight, without mass, would simply be a meaningless number: the direction in this case being partially towards the larger of two massive bodies. (Towards the earth’s centre, in case of weights on Earth.) Quantities like displacement, acceleration and thrust all have directions associated with them; these are all vectors.

Note that as a result of directions coming into the picture, for every vector F, the vector with equal magnitude (see definition below) but going in the exact opposite direction is -F. Vectors are generally represented in bold.

Magnitude

As long as we are talking about terms, let us throw another one in: magnitude. When stating a scalar above, we said 24, when stating our vector we said 80. Irrespective of the units, these numerical values that tell us just how large a scalar or vector is is called its magnitude.

Its companion term, direction, it is worth noting, can be stated in various ways. It could be relative to a compass (north, 17° north-west etc.) or angularly (24° or pi radians) and so on.

There could be possible misunderstanding of magnitudes alongside units, thanks to the evolving definitons of these words. Ideally, when one says 65 metres, the magnitude is 65 and metres is the unit. Much like vector magnitudes and directions, neither of these makes sense without the other.

However, over time, many have come to consider the magnitude and unit together as the magnitude itself, so, to accommodate for that, it becomes important for us to understand that when we say 60m northwest, (northwest being the direction) the whole term of 60m may be considered the magnitude as well.

Types of vectors

There are several types of vectors, but we will talk about the five relatively more basic types: null vectors, position vectors, displacement vectors, projection vectors, and unit vectors.

A null vector has zero magnitude. As for direction, in the strictest terms, these vectors are considered orthogonal to every other vector; but it can also be stated (more conveniently) that null vectors have no direction — although it would then seem to go against the definition of a vector.

A unit vector is like a step up from a null vector in that it has a magnitude of 1. The idea of a unit vector is mainly to incorporate a direction into an existing statement without altering the magnitudes arbitrarily. (Since multiplication with 1 ultimately makes no difference.) The direction of a unit vector is its most important bit. For instance, something may move 1m along a straight line in a particular direction. This statement does not change the existing magnitude (whatever it may be) but gives us directional information. Unless nothing exists, in which case the 1m is itself the magnitude. Generally, the direction along the x-axis is called i, that along the y-axis is called j, and along z is k.

The concept of vector projections is yet another step up. These are more or less just vector resolutions. Generally, vectors are easier to operate on when they lie along an axis (x, y, z etc.) but this great convenience is hardly ever bestowed upon physicists; most vectors are somewhere in-between, so the trick is to write them as a combination of vectors that lie on the axes and then work with them (see fig. 1 above). The scalars ax (length OA) and ay (legth OB) are called the projections of vector a on the x and y axes respectively; ax and ay are the resolved components or the resolutions of vector a.

(The dot in-between is called the dot-product, which we will talk about in the next article. The value |a| refers to just the magnitude, and not direction, of the vector a. We will talk about this further below.)

Sometimes vectors are used to state the position of a point in space. Considering any origin, O, (the point (0,0), (0,0,0) etc. where the basic axes, x, y, and z, meet), the position vector of some point P is simply the straight line from O to P. The representation and magnitude of position vectors is discussed in the next section.

Finally we have the displacement vector, which is simply a position vector, but not from an original reference. The reference, instead, is another point somewhere in space. That is to say, instead of drawing a straight line from origin O to random point P, should we draw one from some point S to point P, we would have a displacement vector.

Representation and magnitude of vectors

Vectors can be represented in various methods. Geometrically, it is just a matter of drawing vectors of relative length and accurate direction. For instance, if 10 m is represented as 2 cm on paper, then a 1 km long vector is represented as 2 m or 200 cm long vector on paper, also inclined at the exact angle as the original.

A more often used method is non-geometric representation and it comes in two styles. Firstly, we can simply state the co-ordinate points. Or we could alternately state it as an equation with x, y and z components on a Cartesian plane (the regular x, y or x, y, z axes grid system everyone is familiar with). There are others, but we will restrict ourselves to the simple Cartesian plane for now.

As co-ordinate points and equations

Suppose we have a point P(x,y,z) somewhere. Then it’s position vector equation would simply be $r = x \hat{i} + y \hat{j} + z \hat{k}$. Conversely, given some equation like $r = 4 \hat{k}$, we can state the point in question as P(0,0,4); or $r = 2 \hat{i} - 5 \hat{j} + 73.8 \hat{k}$ as $P(2,–5,73.8)$.

As for displacement vectors between two points, say Q(a,b,c) and P(x,y,z), we have an $r = (x-a) \hat{i} + (y-b) \hat{j} + (z-c) \hat{k}$ which is the displacement vector. Note how this is really just a generalised form whose specific form was seen above, i.e. with Q(a,b,c) being the origin (0,0,0) we would get the equation we saw eariler as the position vector.

Calculating the magnitude

Knowing a vector $r = a \hat{i} + b \hat{j} + c \hat{k}$, we can calculate its magnitude as the square root of the sum of squares of scalar components. The magnitude, as we saw above, is represented as $|r|$ and in spirit is simply the scalar $r$.

$$ |\textbf{r}| = \sqrt{a^2+b^2+c^2} $$

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‘There are no physicists in the hottest parts of hell, because the existence of a hottest part implies a temperature difference, and any marginally competent physicist would immediately use this to run a heat engine and make some other part of hell comfortably cool. This is obviously impossible.’
— Richard Davisson