Transcript
Vectors
Surveyors use accurate measures of magnitudes and
directions to create scaled maps of large regions.
Vectors
Review:1. What is a scalar?
2. Give some examples of scalars.
3. What is a vector?
4. Give some examples of vectors.
5. How do you represent a vector?
6. What is the denotation (how do you write) of a vector?
Review: Writing a vector1. Common direction
2. Polar coordinate
3. Rectangular coordinate
Review: Resultant
1. What is a resultant?
2. Finding the resultant using graphical solution.
Analytical Vector Resolution:Co-linear Vectors
�⃑� �⃑��⃑�
�⃑� �⃑��⃑�
𝑅=𝐴+𝐵
𝑅=|𝐴−𝐵|
Analytical Vector Resolution (cont…)
1. Triangle method – for adding/subtracting TWO vectors.
The Pythagorean Theorem and SohCahToa:
q�⃑�
�⃑��⃑�
𝑅=√ 𝐴2+𝐵2
𝜃=tan− 1𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
�⃑�
�⃑�
Components of Vectors
A component is the effect of a vector along other directions. The x and y components of the vector are illustrated below.
Finding components:
Polar to Rectangular Conversions
q𝐴𝑥
𝐴𝑦�⃑�
The Pythagorean Theorem and SohCahToa:
𝐴𝑥=𝐴cos𝜃
𝐴𝑦=𝐴 sin 𝜃
𝐴𝑦
�⃑�
�⃑��⃑�
𝑅𝑥=𝐴𝑥+𝐵𝑥+…+𝑛𝑥
𝑅𝑦=𝐴𝑦+𝐵𝑦+…+𝑛𝑦
𝑅=√𝑅𝑥2+𝑅𝑦
2
𝛼=tan−1𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Note: is not yet the direction!!!
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|
Sample Problem:
�⃑� 1=15.0N ,40o N of W
�⃑� 2=25.0N ,180o
�⃑�𝟏− �⃑� 𝟐+ �⃑�𝟑=?�⃑� 3=18.0N ,210
o
Polar Angle x-component y-component
140 -11.5 9.6180 25.0 0210 -15.6 -9.0
-2.1 0.6
�⃑�=2.2N ,164o
𝑅=√𝑅𝑥2+𝑅𝑦
2=√ (−2.1N )2+(0.6N )2=2.2N𝛼=tan−1
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
= tan− 10.62.1
=16o
𝛼
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