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Class 7: Vector and scalar, components Vector operations in - - PowerPoint PPT Presentation
Class 7: Vector and scalar, components Vector operations in - - PowerPoint PPT Presentation
Class 7: Vector and scalar, components Vector operations in components Multiplying a vector with a scalar: y x component y component A+ B A A x A y mA mA x mA y Addition of vectors: x component y component A A x A y B B x B y
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Two common ways to define a vector There are many ways to define a vector, the two most common
- nes are:
x y Ay j i Ax
By components or unit vectors
A = (Ax , Ay ) or A = Ax i + Ay j
By magnitude and angle (direction)
y |A| x
A = (|A| , )
|A| is independent of the choice of the coordinate system. Is usually measured from the +x axis, but not necessary.
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Right angled triangle
If you know two sides, or one side and
- ne angle, you know everything about
the right angled triangle.
A B a b c C
- 2
2 2 2 2
90 B A cos sin side adjacent side
- pposite
tan hypotenuse side adjacent cos hypotenuse side
- pposite
sin 1) cos sin (or c b a
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Conversion between components and magnitude and direction an example
x y 3m
- 4m
vx = ‐4m/s ry = 3m/s or v = (‐4 i + 3 j ) m/s By components or unit vectors:
j i
By magnitude and direction: |v| = 5m/s = 36.87o
- r
|v| = 5m/s = 180o‐ 36.87o = 143.13o
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Same vector in different coordinate axes
x' y' x y 3m
- 4m
20o 5m
Magnitude of a vector is the same in all coordinate system (invariant).
36.87o ?
? = 90o – 36.78o – 20o = 33.22o
rx' = 5 sin 33.22o = 2.739m ry' = - 5 cos 33.22o = - 4.183m r = (2.739 i’ - 4.183 j’) m rx = -4m ry = 3m i' j'
manually!
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Vector operations without a coordinate system
Vector × Vector Complicated. No division between vectors
A 3A ‐A
Multiplying a vector with a scalar
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