10/8 Do now• The diagrams below represent two types motions.

One is constant motion, the other, accelerated motion. Which one is constant motion and which one is accelerated motion? Explain your answer.

A.

B.

Essay homework is dueChapter 3 project – due Tuesday 10/15

Tonight’s Homework – read text book page 84-85 and write an essay to indicate:1. How to distinguish between a scalar and a

vector?2. How is a vector represented?3. How to add and subtract vectors graphically?4. What are some properties of vectors?5. How to multiply or divide a vector by a scalar?Be sure to include definitions of scalar, vector, resultant, Use examples in your essay to clarify ideas.

3-1 introduction to vectors1. Distinguish between a scalar and a vector.2. Add and subtract vectors using the graphical

method.3. Multiply and divide vectors by scalars.

No post session after school today or tomorrow

• Vector: a physical quantity that has both a magnitude and a direction. We use an arrow above the symbol to represent a vector.

• Scalar: a physical quantity that has only a

magnitude but no direction.

A

A

Representing Vectors

• Vectors on paper are simply arrows– Direction represented by the way the ARROW POINTS– Magnitude represented by the ARROW LENGTH

• Examples of Vectors– Displacement– Velocity– Acceleration

Equal vectors: same magnitudeSame direction

Opposite vector: same magnitudeopposite direction

Directions of VectorCompass Point

The direction of a vector is often expressed as an angle of rotation of the vector about its "tail" from east, west, north, or south

20 meters at 10° south of west34 meters at 42° east of north

N

W

S

0°

Directions of VectorReference Vector

Uses due EAST as the 0 degree reference,all other angles are measured from that point

20 meters at 190°34 meters at 48°

0°

90°

180°

270°

Reference vector

Changing Systems• What is the reference vector angle for a vector

that points 50 degrees east of south?

• What is the reference vector angle for a vector that points 20 degrees north of east?

50°

270° + 50° = 320°

20° 20°

Vector Diagrams

1. a scale is clearly listed 2. a vector arrow (with arrowhead) is

drawn in a specified direction. The vector arrow has a head and a tail.

3. the magnitude of the vector is clearly labeled.

tail

head

Vectors can be moved parallel to themselves in a diagram

What we can DO with vectors

• ADD/SUBTRACT with a vector– To produce a NEW VECTOR (RESULTANT)

• MULTIPLY/DIVIDE by a vector or a scalar– To produce a NEW VECTOR or SCALAR

demo

Vector Addition• Two vectors can be added together to determine the

sum (or resultant).

The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together

Two methods for adding vectors

• Graphical method: using a scaled vector diagram – The head-to-tail method (tip to tail)– Parallelogram method

• Mathematical method - Pythagorean theorem and trigonometric methods

A B

A + B = ?

Vector addition: head-to-tail method• A cart is pushed in two directions, as the result, the

cart will move in the resultant direction

=+

A + B = C (Resultant)

A

BC A

B

The resultant is from the first tail to the last head.

• Page 85.

The head-to-tail method (triangle method of addition)

1. Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper.

2. Pick a starting location and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m).

3. Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram.

4. Repeat steps 2 and 3 for all vectors that are to be added 5. Draw the resultant from the tail of the first vector to the

head of the last vector. Label this vector as Resultant or simply R.

6. Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m).

7. Measure the direction of the resultant using the reference counterclockwise convention.

Steps for adding vectors using head and tail method

10/9 do now• Add following vectors using head and tail method to

determine the resultant, use a ruler and a protractor.1. 3 m east, and 4 m south.2. 5 m north and 12 meters west.3. 2 m east, 4 m north and 5 m west.

1. Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper.

2. Pick a starting location and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m).

3. Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram.

4. Repeat steps 2 and 3 for all vectors that are to be added 5. Draw the resultant from the tail of the first vector to the head of the last vector,

with an arrow. Label this vector as Resultant or simply R. 6. Using a ruler, measure the length of the resultant and determine its magnitude by

converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m). 7. Measure the direction of the resultant using the reference counterclockwise

convention.

Essay homework is dueChapter 3 project – due Tuesday 10/15

Tonight’s Homework – castle learning

What is A + B?

A

B

RA

B

R

• A cart is pushed in two directions, as the result, the cart will move in the resultant direction

A

B

Parallelogram method

C

Parallelogram vs. head-to-tail

A

B

A

B

Parallelogram: tail and tail touching, the resultant is the diagonal.

A

B

Head-to-tail: head and tail touching, the resultant is from first tail to last head.

2 heads together

Practice – parallelogram method

• Add following vectors to determine the resultant, use a ruler and a protractor.

1.3 m east, and 4 m south.2.5 m north and 12 meters west.

Vector properties• Vector can be moved parallel to themselves in a diagram.

• Vectors can be added in any order (commutative and associative)

• To subtract a vector, add its opposite.

• Multiplying or dividing vectors by scalars results in vectors with different size, but same direction.

BA

ABBA

)( BABA

A

B

A B

A

B

AB

10/10 Do now: what is the title of this animation:?

Vector subtraction

=- ?

=- B A + B(- )A

= +-

vector addition vs. subtraction

Difference A - B

A

B

Resultant A + B

Equilibrant

• The equilibrant vectors of A and B is the opposite of the resultant of vectors A and B.

• Example:A

B

AB

R

EquilibrantA

BR

Equilibrant

Head to tail

Parallelogram

Do Now: What is 6 + 8 ?

Class work

• Page 87, section review #1-5• Section review work sheet 3-1• Page 113, #1-13

• Homework: castle learning

do now• Write all you know all about vector– Definition:– Examples (3):– Representation:– Ways to add vectors graphically, show sketches to

illustrate your understanding

Do Now: What is 6 + 8 ?