Vector

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Vector Scalar Quantities
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Characteristics of a Scalar Quantity

• Only has magnitude
• Requires 2 things
• 1. A value
• 2. Appropriate units
• Ex. Mass 5kg
• Temp 21 C
• Speed 65 mph

3
Characteristics of a Vector Quantity

• Has magnitude direction
• Requires 3 things
• 1. A value
• 2. Appropriate units
• 3. A direction!
• Ex. Acceleration 9.8 m/s2 down
• Velocity 25 mph West

4
More about Vectors

• A vector is represented on paper by an arrow
• 1. the length represents magnitude
• 2. the arrow faces the direction of motion
• 3. a vector can be picked up and moved on
• the paper as long as the length and
  direction
• its pointing does not change

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Graphical Representation of a Vector

• The goal is to draw a mini version of the vectors
  to give you an accurate picture of the magnitude
  and direction. To do so, you must
• Pick a scale to represent the vectors. Make it
  simple yet appropriate.
• Draw the tip of the vector as an arrow pointing
  in the appropriate direction.
• Use a ruler protractor to draw arrows for
  accuracy. The angle is always measured from the
  horizontal or vertical.

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Understanding Vector Directions
To accurately draw a given vector, start at the
second direction and move the given degrees to
the first direction.
N
30 N of E
E
W
Start on the East origin and turn 30 to the North
S
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Graphical Representation Practice

• 5.0 m/s East
• (suggested scale 1 cm 1 m/s)
• 300 Newtons 60 South of East
• (suggested scale 1 cm 100 N)
• 0.40 m 25 East of North
• (suggested scale 5 cm 0.1 m)

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Graphical Addition of Vectors

• Tip-To-Tail Method
• Pick appropriate scale, write it down.
• Use a ruler protractor, draw 1st vector to
  scale in appropriate direction, label.
• Start at tip of 1st vector, draw 2nd vector to
  scale, label.
• Connect the vectors starting at the tail end of
  the 1st and ending with the tip of the last
  vector. This sum of the original
  vectors, its called the resultant vector.

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Graphical Addition of Vectors (cont.)

• Tip-To-Tail Method
• 5. Measure the magnitude of R.V. with a ruler.
  Use your scale and convert this length to its
  actual amt. and record with units.
• 6. Measure the direction of R.V. with a
  protractor and add this value along with the
  direction after the magnitude.

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Tip-to-Tail Method
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Graphical Addition of Vectors (cont.)
5 Km
Scale 1 Km 1 cm
3 Km
Resultant Vector (red) 6 cm, therefore its 6 km.
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Vector Addition Example 1

• Use a graphical representation to solve the
  following A hiker walks 1 km west, then 2 km
  south, then 3 km west. What is the sum of his
  distance traveled using a graphical
  representation?

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Vector Addition Example 1 (cont.)
Answer ????????
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Vector Addition Example 2

• Use a graphical representation to solve the
  following Another hiker walks 2 km south and 4
  km west. What is the sum of her distance traveled
  using a graphical representation? How does it
  compare to hiker 1?

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Vector Addition Example 2 (cont.)
Answer ????????
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Mathematical Addition of Vectors

• Vectors in the same direction
• Add the 2 magnitudes, keep the direction the
  same.
• Ex.
• 3m E 1m E 4m E

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Mathematical Addition of Vectors

• Vectors in opposite directions
• Subtract the 2 magnitudes, direction is the
• same as the greater vector.
• Ex.
  
• 4m S 2m N
  2m S

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Mathematical Addition of Vectors

• Vectors that meet at 90
• Resultant vector will be hypotenuse of a
• right triangle. Use trig functions and
• Pythagorean Theorem.

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Addition of Vectors (contd.)
Parallelogram Law
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Addition of Vectors (contd.)
Head-to-tail
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Mathematical Subtraction of Vectors

• Subtraction of vectors is actually the addition
  of a negative vector.
• The negative of a vector has the same magnitude,
  but in the 180 opposite direction.
• Ex. 8.0 N due East and 8.0 N due West
• 3.0 m/s 20 S of E and 3.0 m/s 20 N of W

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Subtraction of Vectors (cont.)

• Subtraction used when trying to find a change in
  a quantity.
• Equations to remember
• ?d df di or ?v vf vi
• Therefore, you add the second vector to the
  opposite of the first vector.

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Subtraction of Vectors (cont.)

• Ex. Vector 1 5 km East
• Vector 2 4 km North

5 km W (-v1)
4 km N (v2)
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2D Cartesian Coordinates
Look a two dimensional vector in a 2D Cartesian
Coordinate System








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2D Cartesian Coordinates (contd.)
Y






ay
ax
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Vector Components
Resolving a vector The process of finding
the components of the vector.
Coordinate System
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Component of a vector
The component of the vector along an axis is
its projection along that axis
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Component Method (cont.)

• 4. Add all the X components (Rx)
• 5. Add all the Y components (Ry)
• 6. The magnitude of the Resultant Vector is
• found by using Rx, Ry the Pythagorean
• Theorem
• R2 Rx2 Ry2
• 7. To find direction Tan T Ry / Rx

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Component Method (cont.)

• Ex. 1
• V1 2 m/s 30 N of E
• V2 3 m/s 40 N of W
• Find Magnitude Direction
• Magnitude 2.96 m/s
• Direction 78 N of W

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Component Method (cont.)

• Ex. 2
• F1 37N 54 N of E
• F2 50N 18 N of W
• F3 67 N 4 W of S
• Find Magnitude Direction
• Magnitude 37.3 N
• Direction 35 S of W