Vectors

1
Chapter-3

• Vectors

2

Chapter 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

3
Ch 3-2 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

4
Ch 3-3 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
• sa b b a

5
Ch 3-3 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
• (ab)c a (bc)
• Vector subtraction Vector subtraction is
  obtained by addition of a negative vector

6
Check Point 3-1

• 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-maxab347

a
b
c-max
ii) c-mina-b3-41
a
-b
c-min
7
Ch 3-4 Components of a Vector

• Components of a Vector Projection of a vector on
  an axis
• x-component of vector
• its projection on x-axis
• axa cos?
• y-component of a vector
• Its projection on y-axis
• aya sin?
• Building a vector from its components
• a ?(ax2ay2) tan ? ay/ax

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Check Point 3-2

• 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

9
Ch 3-5 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 ab
• then rx ax bx ry ay by rz axz bz
• r rx i ry j rz k

10
Ch 3-6 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
• aaxi ay j b bxibyj
• rxax bx ry ay by
• r rx i ry j

11
Check Point 3-3

• Ans
• a) ,
• b) , -
• c) Draw d1d2 vector using head-to-tail rule
• Its components are ,

• 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
  d1d2?

12
Ch 3-8 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)

13
Ch 3-8 Multiplication of vectors

• Multiplying a vector by a vector
• i) Scalar Product (Dot Product)
• a.b a(b cos?)b(a cos?)
• (axiayj).(bxibyj)
• axbxayby
• where b cos? is projection of b on a and a
  cos? is projection of a on b

14
Ch 3-8 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.ij.jk.k1 (?0, cos?1)
• is a scalar
• Also dot product of two different unit vectors is
  given by
• i.jj.kk.i 0 (?90, cos?0).

15
Check Point 3-4

• 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.b0 cos? 0 and ?
  cos-1(0)90?
• (b) a.b12, 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)

16
Ch 3-8 Multiplication of vectors

• Multiplying a vector by a vector
• ii) Vector Product (Cross Product)
• c ax b absin?
• c (axiayj)x(bxibyj)
• Direction of c is perpendicular to plane of a
  and b and is given by right hand rule

17
Ch 3-8 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
• ixjk jxk i kxi j
• jxi -k kxj -i ixk-j

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Ch 3-8 Multiplication of vectors

• If aaxi ayj and bbxibyj
• Then c axb
• (axi ayj )x(bxibyj)
• axi x (bxi byj) ayj (bxibyj)
• axbx(i x i ) axby(i x j) aybx(jx i )
  ayby(j x j)
• but ixi0, ixjk jxi-k
• Then caxb (axby-aybx) k

19
Check Point 3-5

• 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 axb0 sin ? 0 and ?
sin-1 (0) 0?, 180? (b) a xb 12, sin ? 1 and
? sin-1(1)90?