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Vector Components
Coordinates

Vectors can be described in terms of coordinates.
• 6.0 km east and 3.4 km south
• 1 N forward, 2 N left, 2 N up

Coordinates are associated with axes in a graph.
y
x = 6.0 m
x
y = -3.4 m
Use of Angles

Find the components of
vector of magnitude 2.0 N at
60° up from the x-axis.

Use trigonometry to convert
vectors into components.
Fy
Fy = (2.0 N) sin(60°)
= 1.7 N
60°
Fx
• x = r cos 
• y = r sin 
Fx = (2.0 N) cos(60°)
= 1.0 N

This is called projection onto
the axes.
Ordered Set


The value of the vector in
each coordinate can be
grouped as a set.
Each element of the set
corresponds to one
coordinate.
• 2-dimensional
• 3-dimensional

The elements, called
components, are scalars, not
vectors.

A  ( Ax , Ay )

A  (4.23, 3.66)

v  (v x , v y , v z )

v  (2.1, 10.5, 3.2)
Component Addition

A vector equation is actually
a set of equations.
• One equation for each
component
• Components can be added
like the vectors themselves

A  (3 N,0 N )  ( Ax , Ay )

B  ( 2 N , 4 N )  ( Bx , B y )
  
C  A B
C x  Ax  Bx  5 N
C y  Ay  B y  4 N

C  (C x , C y )  (5 N,4 N)
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 N
2.1 N
4.1 N
Vector Direction


Vector components can also be used to determine
the direction of a vector.
The tangent of the angle from the x-axis is the ratio of
the y-component divided by the x-component.
4.6 N
2.1 N
  27
4.1 N
tan  
Ay
Ax
Components to Angles

Find the magnitude and
angle of a vector with
components x = -5.0 N, y =
3.3 N.
L x y
2
2
y
x = -5.0 N
2
L x y
2
y = 3.3 N
2
tan   y / x
  tan 1 ( y / x)
L

L = 6.0 N
 = 33o above the
negative x-axis
x
Alternate Axes


Projection works on other
choices for the coordinate
axes.
Other axes may make more
sense for a particular
physics problem.
y’

f
x’
f
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