Given two vectors a and b, the included angle is the angle between these two vectors which is less than or equal to 180 degrees. The dot product can be used to determine the included angle between two vectors. To see how to do this, consider the following picture.
By the law of cosines,

Also from the properties of the dot product,

and so comparing the above two formulas,
 (3.12) 
In words, the dot product of two vectors equals the product of the magnitude of the two vectors multiplied by the cosine of the included angle. Note this gives a geometric description of the dot product which does not depend explicitly on the coordinates of the vectors.
Example 3.2.1 Find the angle between the vectors 2i + j − k and 3i + 4j + k.
The dot product of these two vectors equals 6 + 4 − 1 = 9 and the norms are

and

Now the cosine is known, the angle can be determines by solving the equation cosθ = .72058. This will involve using a calculator or a table of trigonometric functions. The answer is θ = .76616 radians or in terms of degrees, θ = .76616 ×
Example 3.2.2 Let u,v be two vectors whose magnitudes are equal to 3 and 4 respectively and such that if they are placed in standard position with their tails at the origin, the angle between u and the positive x axis equals 30^{∘} and the angle between v and the positive x axis is 30^{∘}. Find u ⋅ v.
From the geometric description of the dot product in 3.12

Observation 3.2.3 Two vectors are said to be perpendicular if the included angle is π∕2 radians (90^{∘}). You can tell if two nonzero vectors are perpendicular by simply taking their dot product. If the answer is zero, this means they are perpendicular because cosθ = 0.
Example 3.2.4 Determine whether the two vectors 2i + j − k and 1i + 3j + 5k are perpendicular.
When you take this dot product you get 2 + 3 − 5 = 0 and so these two are indeed perpendicular.
Definition 3.2.5 When two lines intersect, the angle between the two lines is the smaller of the two angles determined.
Example 3.2.6 Find the angle between the two lines,
These two lines intersect, when t = 0 in the first and t = −1 in the second. It is only a matter of finding the angle between the direction vectors. One angle determined is given by
 (3.13) 
We don’t want this angle because it is obtuse. The angle desired is the acute angle given by

It is obtained by using replacing one of the direction vectors with −1 times it.