CE 201 - Statics

Lecture 4

CARTESIAN VECTORS

We knew how to represent vectors in the form of Cartesian vectors in two dimensions. In this section, we will look into how vectors can be represented in Cartesian vectors form and in three dimensions.

x

y

F

Fx

Fy

i

j

F = Fx i + Fy j

Coordinate System•In this course, the right-handed coordinate system will be used.

•Thumb is directed towards the +ve z-axis

Rectangular Components of a VectorIf F is a vector, then it can be resolved into x, y, and z components depending on the direction of F with respect to x, y, and z axes.

F = Fz + FF = Fx + Fy

then, F = Fx + Fy + Fz (the three components of F)

F

F

Fx

Fy

Fz

x

y

z

Unit Vector

• has a magnitude of 1.• If F is a vector having a magnitude of f 0, then

the unit vector which has the same direction as F is represented by:

• uF = (F / f)

• then, F = f uF

• f defines the magnitude of vector F• uF is dimensionless since F and f have the same

set of units.

F

f

uF

Cartesian Unit Vector

The Cartesian unit vectors i, j, and k are used to designate the directions of x, y, and z axes, respectively.

The sense of the unit vectors will be described by minus (-) or plus (+) signs depending on whether the vectors are pointing along the +ve or –ve x, y, and z axes.

i

j

k

x

y

z

Cartesian Vector Representation

The three components of F can be represented by:

F = Fx i + Fy j + Fz k

F

fFxi

Fyj

Fzk

x

y

z

fx

fy

fz

i

j

k

Magnitude of a Cartesian Vector

f = f2 + fz2

f = fx2 + fy2

then, f = fx2 + fy2 + fz2

Hence, the magnitude of vector F is the square root of the sum of the squares of its components.

Direction of a Cartesian Vector

The orientation of vector F is determined by angles , , and measured between the tail of vector F and the +ve x, y, and z axes, respectively.

Cos () = fx / f

Cos () = fy / f

Cos () = fz / f

, , and are between 0 and 180 F

fxi

fyj

fzk

x

y

z

Another way of determining the direction of F is by forming a unit vector in the direction of F.If, F = fx i +fy j + fz k

uF = F / f = (fx/f) i + (fy/f) j + (fz/f) k

where, f = fx2 + fy2 + fz2

then, uF = cos () i + cos () j + cos () ksince f = fx2 + fy2 + fz2

and uF =1

then, cos2 () + cos2 () + cos2 () = 1if the magnitude and direction of F are known, then

F = f uF

= fcos () i + fcos () j + fcos () k= fx i +fy j + fz k

Addition and Subtraction of Cartesian Vectors

If,

F = Fx i + Fy j + Fz k

H = Hx i + Hy j + Hz k

then,

F + H = R

= (Fx + Hx) i + (Fy + Hy) j + (Fz + Hz) k

F - H = R = (Fx - Hx) i + (Fy - Hy) j + (Fz - Hz) k

F

(fx + hx) (fy + hy)

(fz + hz)

x

y

z

H R

Concurrent Force Systems

In general,

FR = Fx i + Fy j + Fz k

Examples

• Example 2.8• Example 2.9• Example 2.10• Example 2.11• Problem 2.62• Problem 2.69