Vector

Scaler versus Vector Scaler ( 向量 ): : described by magnitude  E.g. length, mass, time, speed, etc Vector( 矢量 ): described by both magnitude and direction  E.g. velocity, force, acceleration, etc Quiz: Temperature is a scaler/vector.

Representing Vector Vector can be referred to as  AB or a or a Two vectors are equal if they have the same magnitude and direction  Magnitudes equal: |a| = |c| or a = c  Direction equal: they are parallel and pointing to the same direction A B AB or a CD or c C D How about these? Are they equal? a b

Opposite Vectors magnitudes are equal, parallel but opposite in sense These two vectors are not equal Actually, they have the relation b = - a a b

Rectangular components of Vector A vector a can be resolved into two rectangular components or x and y components x-component: a x y-component: a y a = [ a x, a y ] y x a ayay axax Ө

Addition of Vectors V1V1 V2V2 V1V1 V2V2 V 1 + V 2 V1V1 V2V2 Method 1 Method 2

Subtraction of Vectors V1V1 V2V2 -V 2 V1V1 V 1 - V 2

Scaling of vectors (Multiply by a constant) V1V1 V1V1 V1V1 2V 1 0.5V 1 -V 1

Class work Given the following vectors V 1 and V 2. Draw on the provided graph paper: V 1 +V 2 V 1 -V 2 2V 1 V1V1 V2V2

Class Work For V 1 given in the previous graph: X-component is _______ Y-component is _______ Magnitude is _______ Angle is _________

Rectangular Form and Polar Form For the previous V 1 Rectangular Form [x, y]: [4, 2] Polar Form r Ө : √  magnitude angle x-component y-component

Polar Form  Rectangular Form Since Therefore: V x = |V| cos Ө V y = |V| sin Ө |V||V| VyVy VxVx Ө magnitude of vector V

Example Find the x-y components of the following vectors A, B & C Given :  |A|=2, Ө A =135 o  |B|=4, Ө B = 30 o  |C|=2, Ө C = 45 o y x A B C ӨAӨA ӨBӨB ӨCӨC

Example (Cont’d) For vector A,  A y = 2 x sin(135 o ) =  2, A x = 2 x cos(135 o ) = -  2 For vector B,  B x = 4 x cos(240 o ) = -4 x cos(60 o ) = -2,  B y = 4 x sin(240 o ) = -4 x sin(60 o ) = -2  3 For vector C,  C y = 2 x sin(-315 o ) = 2 x sin(45 o ) =  2  C x = 2 x cos(-315 o ) = 2 x cos(45 o ) =  2

Example What are the rectangular coordinates of the point P with polar coordinates 8π/6 Solution: use x=r sin Ө and y=r cos Ө y=8sin(π/6)=8(1/2)=4 x=8cos(π/6)=8(  3 /2)=4  3 Hence, the rectangular coordinates are [4  3,4] π/6 8 y x

Class work Find the polar coordinates for the following vectors in rectangular coordinates. V 1 = [1,1] r=____ Ө=_______ V 2 =[-1,1] r=____ Ө=_______ V 3 =[-1,-1] r=____ Ө=_______ V 4 =[1,-1] r=____ Ө=_______

Class work a = [6, -10], r=____ Ө=_______ b = [-6, -10], r=____ Ө=______ c = [-6, 10], r=____ Ө=______ d = [6, 6], r=____ Ө=_______

y =’=’ x 0  ’’ ’’ x y 0  ’’ ’’ x y 0  ’’ ’’ x y 0 or

Unit Vector A vector of length 1 unit is called a unit vector represents a unit vector in the direction of positive x-axis represents a unit vector in the direction of positive y-axis represents a unit vector in the direction of positive z-axis Example on 2-dimensional case: -3 i i 5i5i x y -2 j 4j4j j x y 2i2i

Representing a 2D vector by i and j modulus or magnitude (length or strength) of vector For a 3D vector represented by i, j, and k y x |v| a b i j

Vector addition  If  Then Example:  Given  Then y x a b bxbx byby ayay a+ba+b axax

Vector subtraction  Similarly, for  Then Example  Given  Then y x a b byby byby ayay axax -b-b y x a a+(-b)

Exercise A = 2 i + 3 j, B=  i  j A+B =______________ A  B =_______________ 3A =_________________ |A| = ______________ the modulus of B =______

Example  Given a=7 i +2 j and b=6 i -5 j,  find a+b, a-b and modulus of a+b  Solution

Application of force system Find the x and y components of the resultant forces acting on the particle in the diagram Solution: y x

Scalar Product of Vectors Scalar product, or dot product, of 2 vectors: The result is a scalar   =Angle between the 2 vectors

Example: = _________ =___________ =__________ = ___________ 40 degrees 20 degrees 2 1 a b

For x-y-z coordinates, It can be shown that

Example:  If and  Find, and angle between two vectors  Solution: Notice that a  b = b  a

Properties of scalar product 1. Commutative: 2. Distributive: 3. For two vectors and, and a scalar C, Example: Given A = [1,2], B=[2,-3], C=[-4,5] Find: A  (B+C) = _________ A  B + A  C = _________ 3 A  B = __________ A  (3B) = ___________

If two vectors are perpendicular to each other, then their scalar product is equal to zero. i.e. if then Example  Given and  Show that and are mutually perpendicular  Solution:

Vector Product of Vectors Vector product, or cross product, Defined as The result is a vector Magnitude = Pointing in the direction of is a unit vector perpendicular to the plane containing and in a sense defined by the right-handed screw rule 

Right-Hand-Rule for Cross Product a b a X b

Some properties of vector product  If Ө=0 o, then  If Ө=90 o, then  It can be proven that

Properties of vector product  NOT commutative:  Distributive:  For 3D basic unit vector:  Easy way: using right-hand rule

 For 3D vectors:  By determinant:

Example:  Simplify  Solution:

Example:  Evaluate and calculate if and  Solution: