Points and Vectors

A First Glimpse of Vectors

Before entering into analytic geometry, we observe that R^2, viewed as the set of ordered pairs of real numbers, can be used for other purposes than representing positions of geometric points. For example to describe the current weather, we could agree to write (27,1013) to describe that the temperature is 27 Celsius and the air pressure 1013 millibar. We then describea certain weather situation as an ordered pair of numbers, such as (27,1013).

Of course the order of the two numbers is critical for the interpretation. A weather situation described by the pair (1013,27) with temperature 1013 and pressure 27, is certainly very different from that described by (27,1013) with temperature 27 and pressure 1013.

Having liberated ourselves from the idea that a pair of numbers must represent the coordinates of a point in a Euclidean plane, there are endless possibilities of forming pairs of numbers with the numbers representingdifferent things. Each new interpretation may be viewed as a new interpretation of R^2.

In another example related to the weather, we could agree to write (8,NNE) to describe that the current wind is 8 m/s and headed North-North-East (and coming from South-South-East. Now, NNE is not a real number, so in order to couple to R^2, we replace NNE by the corresponding angle, that is by 22.5^\circ counted positive clockwise starting from theNorth direction. We could thus indicate a particular wind speed and direction by the ordered pair (8,22.5).

You are no doubt familiar with the weather man’s way of visualizing such a wind on the weather map using an arrow.

The wind arrow could also be described in terms of another pair of parameters,namely by how much it extends to the East and to the North respectively, that is by the pair (8\sin(22.5^\circ ),8\cos(22.5^\circ ))\approx (3.06, 7.39). We could say that 3.06 is the “amount of East”, and 7.39 is the “amount of North” of the wind velocity, while we may say that the wind speed is 8, where we think of the speed as the “absolute value” of the wind velocity (3.06, 7.39). We thus think of the wind velocity as having both a direction, and an “absolute value” or “length”.

In this case, we view an ordered pair (a_1,a_2) as a  vector, rather than as a point, and we can then represent the vector by an arrow.

We will soon see that ordered pairs viewed as vectors may be\index{vector} scaled through multiplication by a real number and two vectors may also be added.

Addition of velocity vectors can be experienced on a bike where the wind velocity and our own velocity relative to the ground add together to form the total velocity relative to the surrounding atmosphere, which is reflected in the air resistance we feel.

To compute the total flight time across the Atlantic, the airplane pilot adds the velocity vector of the airplane versus the atmosphere and the velocity of the jet-stream together to obtain the velocity of the airplane vs the ground.

We will return below to applications of analytic geometry to mechanics, including these examples.

Ordered Pairs as Points or Vectors/Arrows

We have seen that we may interpret an ordered pair of real numbers (a_1,a_2) as a  point a in R^2 with coordinates a_1 and a_2. We may write a=(a_1,a_2) for short, and say that a_1 is the first coordinate of the point a and a_2 the second coordinate of a.

We shall also interpret an ordered pair (a_1,a_2)\in R^2 in a alternative way, namely as an arrow with tail at the origin and the head at the point a=(a_1,a_2):

A vector with tail at the origin and the head at the point a=(a_1,a_2)

With the arrow interpretation of (a_1,a_2), we refer to (a_1,a_2) as a vector. Again, we agree to write a=(a_1,a_2), and we say that a_1 and a_2 are the components of the arrow/vector a=(a_1,a_2). We say that a_1 is the first component, occurring in thefirst place and $a_2$ the second component occurring in the second place.

We thus may interpret an ordered pair (a_1,a_2) in R^2 in two ways:

  • as a point with coordinates (a_1,a_2),
  • as an arrow/vector with components (a_1,a_2) starting at the origin and ending at the point (a_1,a_2).

Evidently, there is a very strong connection between the point and arrow interpretations, since the head of the arrow is located at the point (and assuming that the arrow tail is at the origin).

In applications, positions will be connected to the point interpretation and velocities and forces will be connected to the arrow/vector interpretation. We will below generalize the arrow/vector interpretation to include arrows with tails also at other points than the origin.

The context will indicate which interpretation is most appropriate for a given situation. Often the interpretation of a=(a_1,a_2) as a point or as an arrow, changes without notice. So we have to be flexible and use whatever interpretation is most convenient or appropriate. We will need even more fantasywhen we go into applications to mechanics below.

Sometimes vectors like a=(a_1,a_2) are marked by boldface {\bf a} or an arrow \vec{a} or under-score \underbar{a}, or double-script or some other notation. We prefer not to use this more elaborate notation, which makes the writing simpler, but requires fantasy from the user to make the proper interpretation of for example the letter a as a scalar number, or vector a=(a_1,a_2) or something else.

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