Unit Vectors
Vector Length


Vector components can be used to determine the
magnitude of a vector.
The square of the length of the vector is the sum of
the squares of the components.
d  d x2  d y2
4.6 km
2.1 km
4.1 km
Unit Length

A vector with magnitude of exactly 1 has unit length.
• This is vector does not measure units like meters
• Unit vectors have no units!
• The important feature of a unit vector is its direction

A vector can be made by multiplying a scalar
magnitude times a unit vector in the proper direction.
A = 4.6 km
u=1

A  Auˆ
Cartesian Coordinates

A special set of unit vectors
are those that point in the
direction of the coordinate
axes.
• iˆ points in the x-direction.
• ĵ points in the y-direction
• k̂ points in the z-direction
y
ĵ
x
iˆ
Unit Vectors or Components

A vector can be listed in
components.

A  ( Ax , Ay , Az )

A vector’s components can
be used with unit vectors.

A  Axiˆ  Ay ˆj  Az kˆ
Projection


A vector is projected onto each coordinate axis.
The magnitude of the projection is multiplied times a
unit vector.
y

A  Axiˆ  Ay ˆj
Ay ˆj
q
x
Ax iˆ
Projection and Trigonometry



The use of trigonometry can
be combined with the
projections onto the
coordinate axes.
The magnitude of A and the
angle q become
components.
The vector A is represented
by the components and unit
vectors.
Ax  A cos q
Ay  A sin q

A  ( A cos q )iˆ  ( A sin q ) ˆj
Unit Vector Notation

Write the vector of
magnitude 2.0 km at 60° up
from the x-axis in unit vector
notation

Find the components.
y
y = (2.0 km)
sin(60°) = 1.7 km
60°
• x = r cos q = 1.0 km
• y = r sin q = 1.7 km
x
x = (2.0 km)
cos(60°) = 1.0 km

Use unit vectors

A  (1.0 km)iˆ  (1.7 km) ˆj
Alternate Axes


Projection works on other choices for the coordinate
axes.
Other axes may make more sense for the physics
problem.
y’
Ay ˆj

A  Ax iˆ  Ay ˆj 
f
x’
Ax iˆ
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