Phy2 001 vectors

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