1. Engineering Fundamentals Session 6 (1.5 hours)

2. 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.

3. B AB or a CD or c D A C a b 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 How about these? Are they equal?

4. 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

5. a ay Ө ax Rectangular components of Vector y • A vector a can be resolved into two rectangular components or x and y components • x-component: ax • y-component: ay • a = [ax, ay] x

6. Addition of Vectors V1 + V2 V2 V1 V1 V1 V1 + V2 V2 V2 Method 2 Method 1

7. Subtraction of Vectors -V2 V1 V1 - V2 V1 -V2 V2

8. Scaling of vectors (Multiply by a constant) V1 V1 2V1 0.5V1 V1 -V1

9. Class work • Given the following vectors V1 and V2. Draw on the provided graph paper: • V1+V2 • V1-V2 • 2V1 V1 V2

10. Class Work • For V1 given in the previous graph: • X-component is _______ • Y-component is _______ • Magnitude is _______ • Angle is _________

11. Rectangular Form and Polar Form • For the previous V1 • Rectangular Form (x, y): [4, 2] • Polar Form (r, Ө): √20 26.57  or (√20 , 26.57  ) y-component x-component angle magnitude

12. |V| Vy Ө Vx Polar Form  Rectangular Form • Vx = |V| cos Ө • Vy = |V| sin Ө magnitude of vector V

13. y x Example • Find the x-y components of the following vectors A, B & C • Given : • |A|=2, ӨA =135o • |B|=4, ӨB = 30o • |C|=2, ӨC = 45o A ӨA C ӨB ӨC B

14. Example (Cont’d) • For vector A, • Ax=2 x sin(135o)= 2, Ay=2 x cos(135o)=-2 • For vector B, • Bx=4 x sin(210o)= -4 x sin(60o)=-2, • By=4 x cos(210o)= -4 x cos(60o)=-23 • For vector C, • Cx=2 x sin(45o)= 2, Cy=2 x cos(45o)=-2

15. 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)

16. Rectangular Form -> Polar Form • Given (Vx, Vy), Find (r, Ө) • R =  Vx2+Vy2 (Pythagorus Theorm) • Ө = tan -1 (Vx / Vy) ? Will only give answers in Quadrants I and VI • Need to pay attention to what quadrant the vector is in…

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

18. Classwork • Find the polar coordinates for the following vectors in rectangular coordinates. • V1= (1,1) r=____ Ө=_______ • V2=(-1,1) r=____ Ө=_______ • V3=(-1,-1) r=____ Ө=_______ • V4=(1,-1) r=____ Ө=_______

19. Class work • a = (6, -10) r=____ Ө=_______ • b = (-6, -10) r=____ Ө=______ • c = (-6, 10) r=____ Ө=______ • d = (6, 6) r=____ Ө=_______

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