vectors introduction.notebook january 17,...
TRANSCRIPT
Vectors Introduction.notebook
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vectors introduction
Ch.8 Lesson 4
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Target
Agenda
Purpose
Evaluation
Agenda
Purpose
Evaluation
TSWBAT: Determine vector components, Add and subtract vectors, convert to polar form
Warm-Up/Homework Check
Lesson
BAT: understand physics and parametrics
3-2-1
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DefinitionsScalars: single unit 5g or 7in
Vectors: direction and length (magnitude), position doesn't matter
Magnitude: length |AB| is the distance from A to B. ex. |v| =5
Equal Vectors: same direction and magnitude, same displacement different location (helpful when adding or subtracting) a1 = a2 and b1 = b2
Standard position: starting from the origin
AB or v
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Vectors and their Components
horizontal component
vertical component
v = < a , b >
• use two points to find vector components
• A vector is not a specific arrow
• order matters for direction
• start with terminal point
X2 X1 Y2 Y1
(X2 ,Y2)(X1 ,Y1)
Initial Terminal
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Examples
ex. Given points find the vector they create u? Initial: (5,4) Terminal: (4,8)
ex. What are the components for u?
In: (3,3) Ter: (4,4)
ex. If v = <4,7> and the In: (4,5) what is the terminal point?
v = < A , B >X2 X1 Y2 Y1
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Sketch w= <2,3> Int: (0,0), (2,2) and (2,1)
Sketching Examples
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Finding the Magnitude
Def: The magnitude is a single number NOT a vector
If v = <a,b> then the magnitude of
v is |v| = a2 + b2
ex. u = < 12/13, 5/13>, what is the magnitude |u|
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Unit Vectors have magnitude/ length of 1
i = <1,0> and j = <0,1>
ex. check to see if w = < 4/3, 4/5> is a unit vector
Un;t Vectors
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Multiplication by a scalar
when multiplying a vector by a number it stretches or shrinks th magnitdude/length of the vector.
if positive keeps the same direction
BUT if negative it reverses the direction
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Multiplying by a scalar ex.
5v = 5|v|
(1/2)v = (1/2)|v|
(1)v = v
(3)v = 3|v|
stretches magnitude of vector same direction
shrinks magnitude, same direction
reverse direction, same magnitude
stretches magnitude, reverse direction
magnitu
de
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Adding (Sum) of Vectors ‐what it looks like
Def: heading in one direction then heading in another from the endpoint of the first
Algebra: AB + CD = AD
Visual: + =
Reason:
Algebra: CD + AB = AD
Visual: + =
Reason:
Because addition is communitive, the same goes for if the vector order is switched:
d
c a
b
cd
a
12
The vectors AB and BC are the same length. BUT
Why?
Question to think about:
ex. 2
ex.1
u+v
or u+v
b
u v
ab
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Example 2
+ =?
• go along first vector then along the second vector
• Vector sum is long diagonal of parallelogram
+ =dc a
b
c
b
Where will the sum vector be?
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Subtracting ( Difference) of VectorsDef: heading in one direction then heading in another from the endpoint of the first BUT IN THE OPPOSITE DIRECTION
Algebra: AB CD = AD
Visual:+ =
Reason:
Because subtraction is not communitive, the direction is switched, maginitude:
Algebra: BC AB = AC
Visual: + =
Reason:
a
b d
c a d
acbopposite direction
opposite direction
+ =
notice which diagonal it is...short or long?
d
uvu v
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Example 3
=?
A
BB
C
B
CAgo along AB and then backwards BC
short diagonal of parallelogram
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Summarization to write down and understand
B
C
A
B
C AB
B
u + v
Long Diagonal
uv
Short diagonal
if AB is u and BC is v and the angle inbetween is less than 90o
u+vuv
uv
u+v
uv
uv
u + v
Short Diagonal
uv
Long diagonal
if AB is u and BC is v and the angle in between is greater than 90o
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Putting it all together with numbers
u + v = < au + av , bu + bv >
u v = < au av , bu bv >
c v = < c av, c bv >
ex. Find r and s such that
< 3,2> = r < 1,1> + s < 0,4>
ex. Find r and s such that
< 3,2> = r < 1,1> s < 0,4>
Challenge:
ex. is there an <x,y> where there is no r and s such that
<x,y> = r < 1,1> + s < 0,4>
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Another way to write a Vector
in terms of i and j
<a,b> is the same as ai + bj
ex. if u = 3i + 2j and v = i + 6j
find 2u + 5v?
ex. if u = <5,3> and v = <4,6>
find 2u 5v?
ex. <5,8> is the same as 5i 8j
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More we can do with itv= <a,b> v= ai + bj
a = |v| cosθ and b = |v| sinθ
v = |v| cosθ i + |v| sinθ j
Remember |v| = a2 + b2 and
tan1 (b/a) = θ Finding the direction!
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Last examples before application
ex. |v|=8 and direction = 60o
find v in terms of i and j
ex. u= √3 i + j what is the direction?
ex. |v| = 4 and the direction is 30o what is the vector?
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The purpose and application
A pilot is flying on a course of N 42o W at 700 mi/hr with a wind blowing 65 mi/hr toward the south. Find the plane's true velocity, speed, and direction.
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Summarizing
TIME!
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Evaluation:1.
2.
Practice! pg.645 #16,716,18,19, 21,23, 26,28,29,32,34,37
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Warm-Up:
1.
2.
3.