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Vectors
Surveyors use accurate measures of magnitudes and
directions to create scaled maps of large regions.
Vectors
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
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
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
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
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.
++
--
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
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
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
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
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
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
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
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
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
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|
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
