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r r r
PQ = v u− (7.10)

r r 1 r 2 r 2 r r 2
⇒ uv cos( )θ = ( u + v − v u ) (7.11)
−
2
and the dot product can be written as:

r r 1
uv⋅ = ( u + u + v + v − ( v − u ) 2 − ( v − u ) 2 = u v + u v (7.12)
2
2
2
2
2 1 2 1 2 1 1 2 2 11 2 2
In an n-dimensional space, the above expression is generalized to:
r r
uv⋅= u v + u v +…+ u v (7.13)
11 2 2 nn
and the norm square of the vector can be written as the dot product of the
vector with itself; that is,

r r
r 2
u =⋅ = u + u +…+ u 2 (7.14)
2
2
u u
1 2 n
Example 7.3
r
Parallelism and orthogonality of two vectors in a plane. Let the vectors u
r
r
r
ê
and v be given by: u = 3 ê + 4 ê and v = aê + 7 . What is the value of a
1 2 1 2
if the vectors are parallel, and if the vectors are orthogonal?
Solution:
Case 1: If the vectors are parallel, this means that they make the same angle
with the x-axis. The tangent of this angle is equal to the ratio of the vector
x-component to its y-component. This means that:
a = 3 ⇒ a =
7 4 21 4/

Case 2: If the vectors are orthogonal, this means that the angle between them
is 90°, and their dot product will be zero because the cosine for that angle is
zero. This implies that:

3a + 28 = ⇒ = − 28 3/
0
a
Example 7.4
Find the unit vector in 2-D that is perpendicular to the line ax + by + c = 0.

© 2001 by CRC Press LLC