14.2. GEOMETRIC SIGNIFICANCE OF THE DOT PRODUCT 287

Taking square roots of both sides you obtain 14.8.It remains to consider when equality occurs. If either vector equals zero, then that vec-

tor equals zero times the other vector and the claim about when equality occurs is verified.Therefore, it can be assumed both vectors are nonzero. To get equality in the second in-equality above, Theorem 14.1.5 implies one of the vectors must be a multiple of the other.Say b= αa. If α < 0 then equality cannot occur in the first inequality because in this case

(a ·b) = α |a|2 < 0 < |α| |a|2 = |a ·b|

Therefore, α ≥ 0.To get the other form of the triangle inequality, a= a−b+b so

|a|= |a−b+b| ≤ |a−b|+ |b| .

Therefore,|a|− |b| ≤ |a−b| (14.10)

Similarly,|b|− |a| ≤ |b−a|= |a−b| . (14.11)

It follows from 14.10 and 14.11 that 14.9 holds. This is because ||a|− |b|| equals the leftside of either 14.10 or 14.11 and either way, ||a|− |b|| ≤ |a−b| .

14.2 Geometric Significance of the Dot Product14.2.1 The Angle Between Two VectorsGiven two vectors a and b, the included angle is the angle between these two vectors whichis less than or equal to 180 degrees. The dot product can be used to determine the includedangle between two vectors. To see how to do this, consider the following picture.

b

a

a−b

θ

By the law of cosines,

|a−b|2 = |a|2 + |b|2 −2 |a| |b|cosθ .

Also from the properties of the dot product,

|a−b|2 = (a−b) · (a−b) = |a|2 + |b|2 −2a ·b

and so comparing the above two formulas,

a ·b= |a| |b|cosθ . (14.12)

In words, the dot product of two vectors equals the product of the magnitude of the two vec-tors multiplied by the cosine of the included angle. Note this gives a geometric descriptionof the dot product which does not depend explicitly on the coordinates of the vectors.