Engineering Fundamentals Session 6 1 5 hours Scaler

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

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: ax • y-component: ay • a = [ax, ay] y a ay Ө ax x

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

Subtraction of Vectors -V 2 V 1 - V 2 -V 2 V 1

Scaling of vectors (Multiply by a constant) V 1 2 V 1 0. 5 V 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 • 2 V 1 V 2

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] y-component x-component • Polar Form (r, Ө) : √ 20 26. 57 or (√ 20 , 26. 57 ) magnitude angle

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

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

Example (Cont’d) • For vector A, – Ax=2 x sin(135 o)= 2, Ay=2 x cos(135 o)=- 2 • For vector B, – Bx=4 x sin(210 o)= -4 x sin(60 o)=-2, – By=4 x cos(210 o)= -4 x cos(60 o)=-2 3 • For vector C, – Cx=2 x sin(45 o)= 2, Cy=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=8 sin(π/6)=8( 3/2)=4 3; • y=8 cos(π/6)=8(1/2)=4 • Hence, the rectangular coordinates are (4 3, 4)

Rectangular Form -> Polar Form • Given (Vx, Vy), Find (r, Ө) • R = V x 2 + V y 2 (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…

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

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