1stle lecture 02 - r1 units, physical quantities and vectors part 2

8/11/2019 1stLE Lecture 02 - R1 Units, Physical Quantities and Vectors Part 2

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Lecture 1: Units, physical quantitiesand vectors (Part 2)

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Objectives Perform addition and multiplication on

vectors Differentiate scalar product and vector

product

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Blue arrow: resultant vector(sum of two or more vectors)

Addition ofvectors


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Addition ofvectorsHow to find the sum

of three vectors A, Band C? Let us countthe ways!

Option 1: add A andB, to get D, then addC to D to get the R:

Option 2: add B andC, to get E, then add

A to E to get R:

Option 3: add A, Band C, to get Rdirectly:

Option 4: add A, Band C in any orderto get R:

= + + = + + = +

=


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Subtraction of vectorsTo construct the vector difference A B, you can either place thetail of B at the head of A or place 2 vectors A and B head to head

Subtracting B to A is equivalent to adding B to A:

With A and B head to tail, A-B is the vector from thetail of A to the head of -B

With A and B head to head , A-B is the vector from thetail of A to the tail of -B

A + (-B) A - B


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Multiplication of vectors with a scalarMultiplying a vector by a positive and negative scalar

Multiplying a vector by apositive scalar changesthe magnitude of thevector but not its direction

Multiplying a vector by anegative scalar changes

its magnitude andreverses its direction

Note: its important draw vectors in scale (e.g. 1cm:1km)


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A cross-country skier skis 1.00km north and then2.00km east on a horizontal snow field. How far andin what direction is she from the starting point?

Solution 1: (graphical)Drawing the vectordiagram to scale,1division: 1km, bycareful measurement,we found the distance

from starting point toend point about 2.2kmand angle is about 63 o.

Example: Addition of vectors (graphical/analytical)


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Solution 2:(analytical, add 1.00km and 2.00km displacement vectors)

Using Pythagorean theorem, the resultant vector is:

= (1.00 )+(2.00 )=2.24

= .. =63.4 east of north Analytical method yieldsmore accurate results

Example: Addition of vectors (graphical/analytical)


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Addition and subtraction of vectors

Resultant vector: = + Not as simple as addition/subtraction of magnitudesAdd/subtract components having the same unit

vector

EXAMPLE :

= m + m = + + = + +

= + 9


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Products of VectorsScalar (dot) product

= cos = cos = + + Results to a scalar

Dot products of unit vectors can be 0 or 1: =1 = 1

= 1 = =0 = =0 = = 0

Dot of same unit vectors = 1

Dot of different unit vectors = 0

Because the product is a scalar:direction does not matters 10


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Products of VectorsVector (cross) product

= = = sin

Results to a vectorCross product of unit vectors is a unit vector or 0:

=0 = 0 = 0

Cross of same unit vectors = 0

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A helpful way to solve vector (cross) product

= = = =

=

= product is a vector - direction matters!

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= +

= =

=

Perform the following vector operations:

Addition:

Subtraction:

Scalar (dot) product:

Vector (cross) product:

= +

=

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= +

= =

=


Addition:

Subtraction:



=

= +

14

= +

=


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= +

= =

=


Addition:

Subtraction:


= = m

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= +

=

=

= +


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= +

= =

=


Addition:

Subtraction:



= + ( ) = + ( ) = 8m

= +

= = m

=

=

= +


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Seatwork- solve problems in yournotebooks

- write the answers only inyour bluebook- indicate the date

August 12, 20141. Blah?

2. Blah blah!

3. Blah blah blah!

4. Blah blah blah blah!

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A river flows from south to north at 5.0km/h. On thisriver a boat is heading east to west perpendicular to thecurrent at 7.0km/h.1. If you were an eagle hovering at rest over the shore,

draw the 2 vectors corresponding to the motion ofthe rivers current and the boat, then use vectoraddition to draw the resultant vector representing thespeed and direction the boat is traveling?

(Note: drawing must be drawn to scale. If you do not have a ruler, you mayestimate the using the width of your pinky, that is 1 pinky width:1km/h)

2. Using your figure in number 1, estimate how fast and

in what direction (no angle needed) is the boat moving?3-4. Use analytical method to calculate the speed anddirection (include angle) the boat is moving.


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=

=

5. + 6. 7.

9. 10 .

8.

5 - 11: Perform the following vector operations:

11 . ( )

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Seatwork

- seatwork will be checked atthe end of the class- if wrong, write the correctanswer- in checking: place the scoreabove the checkers name - the checker must sign under

his/her name & studentnumber

August 12, 20141. Blah?

2. Blah blah!

3. Blah blah blah!

4. Blah blah blah blah!

Score: 3/4Checked by:(signed)Albert Einstein Jr.

(2013-24601)

X Bleh!!!

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Reminder:

Submit your index cards (3X5) must include picture

Recit 1: FridayCoverage: Lectures 1 and 2 only

(Lecture tomorrow and Thursday will beincluded on next weeks recit)

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1 Att d ill b h k d th it t h t th


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1.Attendance will be checked once the recit teacher enters the room:Case 1: if the number of present students is ODD (if all are present OR 2, 4 , arelate/absent)- have a raffle (using index card) among the present students; 1 lucky raffle winnercan choose which group he/she want to answer the recit for the day, therefore, we

will have 1 group with 3 members; while other groups will have a pairCase 2: if the number of present students is EVEN (if 1, 3 , are late/absent) - no raffle, students can pair among themselves

2. Late students (as penalty) will have to answer the recit quiz on his/her own.Those who come to class AFTER the end of the attendance roll-call will be

considered late. If the student comes before the end of the attendance roll-call,he/she will be accepted and considered NOT LATE even if his/her name wasalready called.

3. Recitation quizzes shall be done in closed notes, students may discuss withtheir partner and encouraged to consult the recit teacher. Students are only

allowed to use real calculators (not the ones from their tablets, laptops orcellphones)

4. Recit solutions will be submitted by group/pair. Write the complete solution toeach problem and box the final answer.

5. If a student complains about the partner, the recit teacher can arrange rotation

partners for the next recit session.


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Seatworks Answers August 8, 2014

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= +12

RHS:

+2= + 2 = Length

LHS:s = Length

LHS and RHS are consistent (unit: Length) 1pt

1: Dimensional Analysis

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2. Bearing form: = 12.0m, 25 o South of West

= . = . = . = .

Seatwork 2 to 5: Write vector andbearing and component form

4. Component form:

= . .

3. Bearing form: = 10.0m, 53 o West of North

= . = . = . = .

5. Component form:

= +

Cy

Dx

25

Dy

Cx


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6 and 7. What is the magnitude anddirection of vector q (include which

quadrant)? = 2.0 +4.

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= + = (2 )+(4 )=4.5 =tan =tan 4 2 = 63.4o or 360 63.4 = 296.6 or 116.6 ANS: 115.6 o in the 2 nd quadrant Any angles that

differ by 180 O have the sametangent

1stle lecture 02 - r1 units, physical quantities and vectors part 2

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