lecture 03 vectors
TRANSCRIPT
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Vectors
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Surveyors use accurate measures of magnitudes and
directions to create scaled maps of large regions.
Vectors
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Identifying Direction
A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.)
A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.)
40 m, 50o N of E
EW
S
N
40 m, 60o N of W40 m, 60o W of S40 m, 60o S of E
Length = 40 m
50o60o
60o60o
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Identifying Direction
Write the angles shown below by using references to east, south, west, north.
Write the angles shown below by using references to east, south, west, north.
EW
S
N45o
EW
N
50o
S
500 S of E500 S of E
450 W of N450 W of N
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Vectors and Polar Coordinates
Polar coordinates (R,q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example.
Polar coordinates (R,q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example.
0o
180o
270o
90o
q0o
180o
270o
90o
R
R is the magnitude and q is the direction.
40 m50o
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Vectors and Polar Coordinates
= 40 m, 50o
= 40 m, 120o = 40 m, 210o
= 40 m, 300o
50o60o
60o60o
0o180o
270o
90o
120o
Polar coordinates (R,q) are given for each of four possible quadrants:Polar coordinates (R,q) are given for each of four possible quadrants:
210o
3000
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Rectangular Coordinates
Right, up = (+,+)
Left, down = (-,-)
(x,y) = (?, ?)
x
y
(+3, +2)
(-2, +3)
(+4, -3)(-1, -3)
Reference is made to x and y axes, with + and - numbers to indicate position in space.
++
--
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Trigonometry Review• Application of Trigonometry to
Vectors
y
x
R
q
y = R sin q y = R sin q
x = R cos qx = R cos q
siny
R
cosx
R
tany
x R2 = x2 +
y2
R2 = x2 + y2
Trigonometry
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Finding Components of VectorsA component is the effect of a vector along other directions. The x and y components of the vector are illustrated below.
q
= A cos q
Finding components:
Polar to Rectangular Conversions
= A sin q
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Example 2: A person walks 400.0 m in a direction of 30.0o S of W (210o). How far is the displacement west and how far south?
400 m
30o
The y-component (S) is opposite:
The x-component (W) is adjacent: = -A cos q
= -A sin q
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Vector Addition
Resultant ( )
- sum of two or more vectors.
Vector Resolutions:
01. GRAPHICAL SOLUTION
– use ruler and protractor to draw and measure the scaled magnitude and angle (direction), respectively.
02. ANALYTICAL SOLUTION
- use trigonometry
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Example 11: A bike travels 20 m, E then 40 m at 60o N of W, and finally 30 m at 210o. What is the resultant displacement graphically?
60o
30o
R
fq
Graphically, we use ruler and protractor to draw components, then measure the Resultant R,q
A = 20 m, E
B = 40 m
C = 30 m
R = (32.6 m, 143.0o)
R = (32.6 m, 143.0o)
Let 1 cm = 10 m
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A Graphical Understanding of the Components and of the Resultant is given below:
60o
30o
R
fq
Note: Rx = Ax + Bx + Cx
Ax
B
Bx
Rx
A
C
Cx
Ry = Ay + By + Cy
0
Ry
By
Cy
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Resultant of Perpendicular VectorsFinding resultant of two perpendicular vectors is like changing from rectangular to polar coord.
R is always positive; q is from + x axis
2 2R x y
tany
x x
yR
q
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Vector DifferenceFor vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.
First Consider A + B Graphically:
B
A
BR = A + B
R
AB
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Vector DifferenceFor vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.
Now A – B: First change sign (direction) of B, then add the
negative vector.B
A
B -B
A
-BR’
A
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Comparison of addition and subtraction of B
B
A
B
Addition and Subtraction
R = A + B
R
AB -BR’
AR’ = A - B
Subtraction results in a significant difference both in the magnitude and the direction of the resultant vector. |(A – B)| = |A| - |B|
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Example 13. Given A = 2.4 km, N and B = 7.8 km, N: find A – B and B – A.
A 2.43
N
B 7.74
N
A – B; B -
A
A - B
+A
-B
(2.43 N – 7.74 S)
5.31 km, S
B - A
+B-A
(7.74 N – 2.43 S)
5.31 km, N
R R