(6) The scalar or dot product

There are two different ways in which we can usefully define the multiplication of two vectors. The first of these is called the scalar or dot product.
The picture below gives its definition for two vectors a and b and also shows a very neat practical application.
showing how the dot product works

The man is pulling the block with a constant force a so that it moves along the horizontal ground. The work done in moving the block through a distance b is then given by the distance moved through multiplied by the magnitude of the component of the force in the direction of motion.
This is |a| |b| cos t so we define the scalar or dot product as
a.b = |a||b| cos t

where t is the angle between a and b when they are placed tail to tail.
To use the least amount of force possible, we would need to pull horizontally, so that we are pulling in the same direction as we want the object to move. Then we would have t=0 and cos t=1 so that
work done = a.b = |a||b| = magnitude of the force x distance moved in the direction of the force.

IMPORTANT Each of the lengths |a| and |b| is a number and cos t is a number, so a.b is not a vector but a number or scalar. This is why it's called the scalar product. When writing down two vectors multiplied in this way, you must include the dot between them. Writing ab is meaningless.
(If you are visiting from can we multiply 3 vectors together? you can return here.)

If the force acting on a particle is varying in magnitude and direction, so that the path moved through by the particle is wiggly, it is still possible to use the dot product to find the work done to move the particle along this path. We do this by adding up the work done over separate very short sections of the path (so short that they are almost straight and the force hardly varies). The shorter we make the sections, the closer the sum becomes to the exact work done, and we finally get what is called a path integral. This very useful extension of the dot product is covered in courses on vector calculus.

Special cases of the dot product

Since i and j and k are all one unit in length and they are all mutually perpendicular, we have
i.i = j.j = k.k = 1
and i.j = j.i = i.k = k.i = j.k = k.j = 0.

(If you are visiting from the scalar triple product, you can return here.)

We now have a neat way of finding whether two vectors are perpendicular.
Because algebra is difficult to show on the web, I'll give a numerical example.

Suppose that a = 3i + 5j - 2k and b = 2i - 2j - 2k.
Then a.b = (3i + 5j - 2k).(2i - 2j - 2k).

It can be shown that it is all right to multiply the two vectors a and b by working out all the separate multiplications of the components. Using this and the results above, we get
a.b = (3 x 2) + (5 x -2) + (-2 x -2) = 6 - 10 + 4 = 0

Since neither |a| nor |b| = 0 we have cos t = 0 so t = 90 degrees and a and b are perpendicular.
(If you are visiting from equations of planes using normal vectors, you can return here.)
We've now got a way of working out the angle between the directions of any two vectors.
(To picture this, you can shift them if necessary so that their tails meet.)
Again I'll take a numerical example. We'll find the angle between
a = 2i - 3j + k and b = 4i + j - 3k.

Using section (5), |a| = the square root of (4 + 9 + 1) = 3.742 to 3 d.p
and |b| = the square root of (16 + 1 + 9) = 5.099 to 3 d.p.
Also we have a.b = (2 x 4) + (-3 x 1) + (1 x -3) = 2.
Since a.b = |a||b| cos t, we now have
2 = 3.742 x 5.099 x cos t so cos t = 0.1048 and t = 84 degrees
measuring to the nearest degree.
You can return here if you are visiting from the angles between lines and planes.

A practical application

a right whale fluke
A right whale fluke © the New England Aquarium

A reader has used the dot product to help him analyse the movements of a tagged right whale. He had the x and y coordinates for a set of positions of the whale and wanted to calculate the angles turned through between successive sections of its journey. He found each section as a vector by calculating the differences between the pairs of x and y coordinates of the endpoints. Then he used the dot product of each successive pair of vectors to find the angle between those two legs of the whale's journey.

There is interesting information about this and related topics at the GIS lab at the New England Aquarium in Boston, Massachusetts.

We can now use exactly the same method to find the three angles which give the direction of a vector relative to the three axes.


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