- Definition
- Finding the angle between two vectors
- Projection
- Proving the cosine formula
- Great circle distances

The dot product of ** a** and

where a and b are the magnitudes of

0 O q O p

The dot product is distributive:

and commutative:

Knowing that the angles between each of the

then,

=>

The angle between any nonzero vector and iteself is 0, and cos 0 = 1, so

This means that for any vector,

We can now, given the coordinates of any two nonzero vectors __ u__ and

=> u v cos q = a x + b y + c z

=> q = cos

What would happen if one of the vectors was the null vector

One of the main uses of the dot product is to determine whether two vectors,

If

It will often be useful to find the component of one vector in the direction of another:

We have a given vector ** a**, and we want to see how far it extends in a direction given by the unit vector

d = a cos q

=> d = n a cos q

=> d =

You have two sides of a triangle, a and b, and the angle in between, C, - the problem is to find the remaining side c. You kill the problem by recalling the cosine formula:

c^{2} = a^{2} + b^{2} - 2 a b cos C

but have you ever seen a proof? The proof by geometry isn't very friendly but with vectors it takes all of 3 lines (using the second triangle above):
** c**.

=> c

=> c

From the latitudes and longitudes of two places on the Earth together with the radius of the Earth we can determine the position vectors of the two places with the origin at the centre of the Earth. If you have two points on the circumference of a circle then the radius of the circle times the angle (in radians) subtended by the two points at the centre of the circle gives the arc distance between the two points. Using the dot product we can find the angle subtended by our two position vectors, multiply by the radius of the Earth, and hey presto we have the great circle distance.

Find out the distance between us using this applet (I'm at latitude 53, longitude 0).