1. Lecture 2
Vectors

2. Agenda Chapter 2 Vector s Addition subtraction components
scalar product
cross product 1

3. vectors
In physics we have Phys. quantities that can be completely described by a number and are known as scalars. Temperature and mass are good examples of scalars.
Other physical quantities require additional information about direction and are known as vectors. Examples of vectors are displacement, velocity, and acceleration.
In this chapter we learn the basic mathematical language to describe vectors. In particular we will learn the following:
Geometric vector addition and subtraction Resolving a vector into its components The notation of a unit vector Addition and subtraction vectors by components Multiplication of a vector by a scalar The scalar (dot) product of two vectors The vector (cross) product of two vectors

4. Vectors and Scalars
Vectors : Vector quantity has magnitude and direction
Vector represented by arrows with length equal to vector magnitude and arrow direction giving the vector direction
Example: Displacement Vector
Scalar : Scalar quantity with magnitude only.
Example: Temperature,mass

5. Adding Vectors Geometrically
Vector addition:
Resultant vector is vector sum of two vectors
Head to tail rule : vector sum of two vectors a and b can be obtained by joining head of a vector with the tail of b vector. The sum of the two vectors is the vector s joining tail of a to head of b
s=a + b = b + a

6. Adding Vectors Geometrically
Commutative Law: Order of addition of the vectors does not matter
a + b = b + a
Associative Law: More than two vectors can be grouped in any order for addition
(a+b)+c= a + (b+c)
Vector subtraction: Vector subtraction is obtained by addition of a negative vector

7. Exemple
The magnitude of displacement a and b are 3 m and 4 m respectively. Considering various orientation of a and b, what is
i) maximum magnitude for c and ii) the minimum possible magnitude?
i) c-max=a+b=3+4=7 a b
c-max
ii) c-min=a-b=3-4=1 a -b
c-min

8. Components of a Vector
Components of a Vector: Projection of a vector on an axis
x-component of vector:
its projection on x-axis
ax=a cos 
y-component of a vector:
Its projection on y-axis
ay=a sin
Building a vector from its components
a =(ax2+ay2); tan  =ay/ax

9. Exemple Ans: Components must be connected following head-to-tail rule.
In the figure, which of the indicated method for combining the x and y components of the vector d are propoer to determine that vector?
Ans:
Components must be connected following head-to-tail rule.
c, d and f configuration

10. ax , ay and az are scalar components of the vector
Unit Vectors
Unit vector: a vector having a magnitude of 1 and pointing in a specific direction
In right-handed coordinate system, unit vector i along positive x-axis, j along positive y-axis and k along positive z-axis.
a = ax i + ay j + az k
ax , ay and az are scalar components of the vector
Adding vector by components: r= a+b
then rx= ax + bx ; ry= ay + by ; rz= axz+ bz
r = rx i+ ry j + rz k

11. Adding Vectors by components
To add vectors a and b we must: 1) Resolve the vectors into their scalar components 2) Combine theses scalar components , axis by axis, to get the components of the sum vector r 3) Combine the components of r to get the vector r r= a + b a=axi + ay j; ; b = bxi+byj rx=ax + bx; ry = ay + by r= rx i + ry j

12. Exemple
a) In the figure here, what are the signs of the x components of d1 and d2? b) What are the signs of the y components of d1 and d2? c) What are the signs of x and y components of d1+d2?
Ans: a) +, + b) +, - c) Draw d1+d2 vector using head-to-tail rule Its components are +, +

13. Multiplication of vectors
Multiplying a vector by a scalar:
In multiplying a vector a by a scalar s, we get the product vector sa with magnitude sa in the direction of a ( positive s) or opposite to direction of a ( negative s)

14. Multiplication of vectors
Multiplying a vector by a vector:
i) Scalar Product (Dot Product)
a.b= a(b cos)=b(a cos)
= (axi+ayj).(bxi+byj)
= axbx+ayby
where b cos is projection of b on a and a cos is projection of a on b

15. Multiplication of vectors
Since a.b= ab cos
Then dot product of two similar unit vectors i or j or k is given by :
i.i=j.j=k.k=1 (=0, cos=1)
is a scalar
Also dot product of two different unit vectors is given by:
i.j=j.k=k.i =0 (=90, cos=0).

16. Exemple
Vectors C and D have magnitudes of 3 units and 4 units, respectively. What is the angle between the direction of C and D if C.D equals:
a) Zero
b) 12 units
c) -12 units?
a) Since a.b= ab cos and a.b=0 cos =0 and = cos-1(0)=90◦;
(b) a.b=12, cos =1 and
= cos-1(1)=0◦
(vectors are parallel and in the same direction)
(c) b) a.b=-12, cos =-1 and
= cos-1(-1)=180◦
(vectors are in opposite directions)

17. Multiplication of vectors
Multiplying a vector by a vector:
ii) Vector Product (Cross Product)
c= ax b = ab sin
c = (axi+ayj) x (bxi+byj)
Direction of c is perpendicular to plane of a and b and is given by right hand rule

18. Multiplication of vectors
Since a x b= ab sin  is a vector
Then cross product of two similar unit vectors i or j or k is given by :
ixi= jxj = kxk =0 (as =0 so sin =0).
Also cross product of two different unit vectors is given by:
ixj=k; jxk= i ; kxi =j
jxi= -k; kxj= -i ; ixk=-j

19. Multiplication of vectors
If a=axi +ayj and b=bxi+byj
Then c = axb
=(axi +ayj ) x (bxi+byj)
= axi x (bxi +byj) + ayj (bxi+byj)
= axbx(i x i ) + axby(i x j) + aybx(jx i ) + ayby(j x j)
but ixi=0, ixj=k; jxi=-k
Then c=axb = (axby-aybx) k

20. Exemple a) Zero b) 12 units
Vectors C and D have magnitudes of 3 units and 4 units, respectively. What is the angle between the direction of C and D if magnitude of C x D equals:
a) Zero
b) 12 units
a) Since a xb= ab sin and axb=0 sin  =0 and = sin-1 (0) =0◦, 180◦
(b) a xb =12, sin  =1 and
= sin-1(1)=90◦