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4 Unit Vector (1.4S)

A vector A has both magnitude and direction. The magnitude of vector A is a scalarquantity written as A or |A|. A unit vector aA along A is defined as a vector that has thesame direction as Abut a magnitude of 1.

aA= A

|A|=

A

A (1)

Note that |aA|= 1. Vector Acan be written as,

A= AaA (2)

5 Representation of Vectors in Cartesian CoordinateSystem

A vector A in Cartesian coordinate system is written as,

(Ax, Ay, Az) or A= Axx+Ayy+Az z (3)

Ax, Ay and Az are called components of A in x, y and zdirections, respectively. The unitvectors x, y and zpoint in the increasing direction ofx, y, and z.

Figure 1: Representation of a vector in Cartesian coordinates [1, fig. 1.1]

The magnitude ofA is given by,

|A|= A =A2x+ A

2y+ A

2z (4)

The unit vector along A is given by,

aA =A

A =

Axx+Ayy+Az zA2x+A

2y+ A

2z

(5)

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6 Addition and Subtraction of Vectors (1.5S)

Vectors Aand B can be added together to give a third vector C,

C= A + B (6)

IfA= (Ax, Ay, Az) and B= (Bx, By, Bz) then,

C= (Ax+ Bx)x+ (Ay+ By)y+ (Az+ Bz)z (7)

Vector subtraction is carried out in a similar manner.

D= A B= A + (B) = (AxBx)x+ (Ay By)y+ (Az Bz)z (8)

Graphically, vectors are added or subtracted using the parallelogram rule or headtotailrule.

7 Function

Warning: This is a very loose definition of function but one that is convenient and suitablefor us. In calculus you might come across a stricter definition.

A function is a rule (or relationship) between two quantities.

Two variables, independent and dependent are associated with a function.

All possible values that independent variable can take is called domain of function.

All possible values that dependent variable can take is called range of function.

In the function x(t) = 2t, x is dependent variable, t is independent variable and therule is value ofx is twice that oft or simply 2t.

8 Scalar and Vector Fields

8.1 Field

A field is a function of space or the coordinates x, y and z. A field specifies or defines ascalar quantity in the whole space or region.

8.2 Scalar Field

A scalar field is a function that assigns to each point in space a scalar or number. For example,temperature in a room at each point is defined by a function T(x,y,z) = 2x+ 3yz.

8.3 Vector Field

A vector field is a function that assigns to each point in space a vector. For example, electric

field in a region is given by E(x,y,z) = 2yx+ 3zy+ xyz.

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9 Exercises

Question 1: Given A= 10x

4y+ 6zand B= 2x+ y, find,

(1) The scalar component ofA along y.

(2) The vector component ofAalong y.

(3) The magnitude of 3A 4B.

(4) A unit vector along A + 2B

Question 2: Given vectors A= x+ 3zand B= 5x+ 2y6z, find,

(1) |A + B|.

(2) 5A B.

(3) The scalar and vector components ofA along y.

(4) A unit vector along (or parallel to) 3A + B.

Question 3: An airplane has a ground speed of 350 km/hr in the direction due west. Ifthere is a wind blowing northwest at 40 km/hr calculate the true air speed and heading orplane.

References

[1] M. N. O. Sadiku. Elements of Electromagnetics. 2nd edition.

Prepared By: Attique Dawood Last Modified By: Arshad Hassan

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