Norm of a Vector

1
Section 3.2

• Norm of a Vector
• Vector Arithmetic

2
PROPERTIES OF VECTOR ARITHMETIC
Theorem 3.2.1 If u, v, and w are vectors in 2-
or 3-space and k and l are scalars, then the
following relationships hold. (a) u v v
u (b) (u v) w u (v w) (c) u 0 0 u
u (d) u (-u) 0 (e) k(lu) (kl)u (f) k(u 
v) ku kv (g) (k l)u ku lu (h) 1u u
3
NORM OF A VECTOR
The length of a vector u is often called the norm
of u and is denoted by u. By the Pythagorean
Theorem, we have
4
DISTANCE BETWEEN POINTS
If P1(x1, y1, z1) and P2(x2, y2, z2) are two
points in 3-space, then the distance between them
is the norm of the vector . That is,
A similar result holds for the distance
between two points in 2-space.
5
REMARKS ABOUT THE NORM

• A vector that has norm 1 is called a unit vector.
• From the definition of ku and the definition of
  norm, we have
• ku k u