The picture below gives its definition for two vectors

The man is pulling the block with a constant force

This is |

where t is the angle between

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

IMPORTANT Each of the lengths

(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.

Since

and

(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.

Then

It can be shown that it is all right to multiply the two vectors

Since neither

(If you are visiting from equations of planes using normal vectors, you can return here.)

We've now got a way of working out

(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

Using section (5),

and

Also we have

Since

measuring to the nearest degree.

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.

or ** back to the previous section **

or **return to the vector homepage.**