(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.
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
a.b = |a||b| cos t
where t is the angle between a and b when they are placed 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.
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.
Suppose that a = 3i + 5j - 2k and
b = 2i - 2j - 2k.
Because algebra is difficult to show on the web, I'll give a numerical
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
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 © 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,
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.
or back to the previous section
or return to the vector homepage.