Chapter-3

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 aand bcan 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

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 (a+b)+c= a + (b+c) • Vector subtraction: Vector subtraction is obtained by addition of a negative vector

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

8. 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 Check Point 3-2

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= a+b 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 a=axi + ayj; ; b = bxi+byj rx=ax + bx; ry = ay + by r= rxi + ryj

11. 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 +, + Check Point 3-3

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) = (axi+ayj).(bxi+byj) = axbx+ayby 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.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).

15. 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) Check Point 3-4

16. Ch 3-8 Multiplication of vectors • Multiplying a vector by a vector: ii) Vector Product (Cross Product) c= ax b = absin c = (axi+ayj)x(bxi+byj) 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: ixj=k; jxk= i ; kxi =j jxi= -k; kxj= -i ; ixk=-j

18. Ch 3-8 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

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 axb=0 sin  =0 and = sin-1 (0) =0◦, 180◦ (b) a xb =12, sin  =1 and = sin-1(1)=90◦