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’
Ay ˆj
A Ax iˆ Ay ˆj
f
x’
Ax iˆ
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