Engineering Fundamentals Session 6 (1.5 hours)

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 ) : √  or (√20,  ) magnitude angle x-component y-component

Polar Form  Rectangular Form 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 x =2 x sin(135 o )=  2, A y =2 x cos(135 o )=-  2 For vector B, –B x =4 x sin(210 o )= -4 x sin(60 o )=-2, –B y =4 x cos(210 o )= -4 x cos(60 o )=-2  3 For vector C, –C x =2 x sin(45 o )=  2, C y =2 x cos(45 o )=-  2

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

Rectangular Form -> Polar Form Given (V x, V y ), Find (r, Ө ) R =  V x 2 + V y 2 (Pythagorus Theorm) Ө = tan -1 (V x / V y ) ? Will only give answers in Quadrants I and VI Need to pay attention to what quadrant the vector is in…

How to Find Angle? Find the positive angle Ø = tan -1 (|V y |/|V x |) Ө = Ø or 180-Ø or 180+Ø or –Ø, depending on what quadrant. Absolute value (remove the negative if any) Ø Ø Ø Ø Ø 180- Ø 180+ Ø -Ø-Ø

Classwork 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 =_______

Concept Map Vectors Polar Form (r, Ө ) Rectangular Form (V x,V y ) operations Addition + Subtraction- Scalar multiplication representation notation V V AB magnitude Angle or phase 2 V V1 – V2V1 – V2 V 1 + V 2 conversion Beware of the quadrant, and use of tan -1 !!!