This document discusses Cartesian vectors and coordinate systems. It defines a right-handed coordinate system and explains how to represent a vector A using its rectangular components along the x, y, and z axes. It also describes how to calculate the magnitude and direction of a Cartesian vector using direction cosines and unit vectors.

1. 2.5 Cartesian Vectors
Right-Handed Coordinate System
A rectangular or Cartesian coordinate
system is said to be right-handed
provided:
- Thumb of right hand points
in the direction of the positive
z axis when the right-hand
fingers are curled about this
axis and directed from the
positive x towards the positive y axis

2. 2.5 Cartesian Vectors
Right-Handed Coordinate System
- z-axis for the 2D problem would be
perpendicular, directed out of the page.

3. 2.5 Cartesian Vectors
Rectangular Components of a Vector
- A vector A may have one, two or three
rectangular components along the x, y and z
axes, depending on orientation
- By two successive application of the
parallelogram law
A = A’ + Az
A’ = Ax + Ay
- Combing the equations, A can be
expressed as
A = Ax + Ay + Az

4. 2.5 Cartesian Vectors
Unit Vector
- Direction of A can be specified using a unit
vector
- Unit vector has a magnitude of 1
- If A is a vector having a magnitude of A ≠ 0,
unit vector having the same direction as A is
expressed by
uA = A / A
So that
A = A uA

5. 2.5 Cartesian Vectors
Unit Vector
- Since A is of a certain type, like force
vector, a proper set of units are used for the
description
- Magnitude A has the same sets of units,
hence unit vector is dimensionless
- A ( a positive scalar)
defines magnitude of A
- uA defines the direction
and sense of A

6. 2.5 Cartesian Vectors
Cartesian Unit Vectors
- Cartesian unit vectors, i, j and k are used
to designate the directions of z, y and z axes
- Sense (or arrowhead) of these
vectors are described by a plus
or minus sign (depending on
pointing towards the positive
or negative axes)

7. 2.5 Cartesian Vectors
Cartesian Vector Representations
- Three components of A act in the positive i,
j and k directions
A = Axi + Ayj + AZk
*Note the magnitude and
direction of each components
are separated, easing vector
algebraic operations.

8. 2.5 Cartesian Vectors
Magnitude of a Cartesian Vector
- From the colored triangle,
A = A'2 + Az2
- From the shaded triangle,
A' = Ax2 + Ay
2
- Combining the equations gives
magnitude of A
A = Ax2 + Ay + Az2
2

9. 2.5 Cartesian Vectors
Direction of a Cartesian Vector
- Orientation of A is defined as the
coordinate direction angles α, β and γ
measured between the tail of A and the
positive x, y and z axes
- 0° ≤ α, β and γ ≤ 180 °

10. 2.5 Cartesian Vectors
Direction of a Cartesian Vector
- For angles α, β and γ (blue colored
triangles), we calculate the direction
cosines of A
Ax
cos α =
A

11. 2.5 Cartesian Vectors
Direction of a Cartesian Vector
- For angles α, β and γ (blue colored
triangles), we calculate the direction
cosines of A
Ay
cos β =
A

12. 2.5 Cartesian Vectors
Direction of a Cartesian Vector
- For angles α, β and γ (blue colored
triangles), we calculate the direction
cosines of A
Az
cos γ =
A

13. 2.5 Cartesian Vectors
Direction of a Cartesian Vector
- Angles α, β and γ can be determined by the
inverse cosines
- Given
A = Axi + Ayj + AZk
- then,
uA = A /A
= (Ax/A)i + (Ay/A)j + (AZ/A)k
where A = Ax2 + Ay + Az2
2

14. 2.5 Cartesian Vectors
Direction of a Cartesian Vector
- uA can also be expressed as
uA = cosαi + cosβj + cosγk
- Since A = Ax + Ay + Az2 and magnitude of uA
2 2
= 1,
cos 2 α + cos 2 β + cos 2 γ = 1
- A as expressed in Cartesian vector form
A = AuA
= Acosαi + Acosβj + Acosγk
= Axi + Ayj + AZk