Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such...
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![Page 1: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/1.jpg)
Vectors in Two Dimensions
![Page 2: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/2.jpg)
VECTOR REPRESENTATIONA vector represents those physical quantities such as velocity that have both a magnitude and a direction.Vectors can be represented as arrows whose length is proportional to the vector's magnitude and whose direction is the vector's direction.Manipulation of vector's can be greatly simplified when they are displayed on a Cartesian Coordinate System. This allows the vector to be manipulated in both Cartesian and Polar coordinates.One of the major simplifications comes from representing a vector in terms of its components along the x and y axes. The simplification comes from the fact that these components are scalar quantities that may be manipulated according to the rules of algebra. Of course we often want to take the components of a vector and represent the vector itself.Vectors may be added, subtracted, multiplied by a scalar, and multiplied by other vectors.
![Page 3: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/3.jpg)
When representing a vector on a Cartesian Coordinate System, the x and y coordinates of the point of the arrow will be equal to the vector's x and y components. The direction of the vector will be the angle measured counterclockwise from the +x axes.
+x-x
+y
-y
origin
r V
V
θVx =V cosθ
VY =V sinθ
Vector Magnitude
Vector Direction
x-component
y-component
![Page 4: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/4.jpg)
+x-x
+y
-y
Quadrant I0° < θ < 90°
V X > 0, V Y > 0
Quadrant III180° < θ < 270°V X < 0, V Y < 0
Quadrant IV270° < θ < 360°V X > 0, V Y < 0
Quadrant II90° < θ <180°V X < 0, V Y > 0
![Page 5: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/5.jpg)
Determining a Vector from its Components
VY
VX
V
θ
V =V = V X2 + V Y
2
θ =tan−1 VYVX
⎛ ⎝ ⎜ ⎞
⎠ ⎟
r V
r V =V @ θ
![Page 6: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/6.jpg)
For Vectors in Quadrants II, III, IV III
III IV
V =V = V X2 + V Y
2
θ =tan−1 VYVX
⎛ ⎝ ⎜ ⎞
⎠ ⎟
V =V = V X2 + V Y
2
V =V = V X2 + V Y
2 V =V = V X2 + V Y
2
θ =180° − tan−1 Vy
Vx
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
θ =180° + tan−1 Vy
Vx
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ θ =360° − tan−1 Vy
Vx
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
![Page 7: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/7.jpg)
Practice1. Find the components of the vector V= 15@130°
15130°
Vx =V cosθ
Vx =15⋅cos130°
Vx =−9.64
Vx
Vy =V sinθ
Vy =15⋅sin130°
Vy =11.5
Vy
![Page 8: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/8.jpg)
2. Find the vector with the following x and y components:Vy =−12.8Vx =−8.1
Vx =−8.1
V y=−12.8V
θ
V = V x2 +V y
2
V = −8.1( )2 + −12.8( )2
V =15.1
θ =180° + tan−1 Vy
Vx
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
θ =180° + tan−1 − 12.8− 8.1
⎛ ⎝ ⎜ ⎞
⎠ ⎟
θ =180° + tan−1 1.58( )θ =180° + 57.7°
θ =237.7°
r V =15.1@ 237.7°
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Operations with
Vectors
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Adding and Subtracting VectorsThe sum of or difference between two or more vectors is a new vector.
r R =
r A +
r B +
r C
r R =R @ θ
R = R x2 +R y
2
θ
tan−1R yR x
⎛ ⎝ ⎜
⎞ ⎠ ⎟L Q : I
180 −tan−1R yR x
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟L Q : II
180 + tan−1R yR x
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟L Q : III
360 −tan−1R yR x
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟L Q : IV
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
Where :Rx =Ax +Bx +Cx
R y =Ay +By +Cy
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Multiplying a Vector by a ScalarThe product of a vector and a positive scalar is a new vector in the same direction as the original vector but with a new magnitude.
r V =V @ θ
r R =s
r V
r R =sV@ θK s> 0
Rx =sVxR y =sVy
The product of a vector and a negative scalar is a new vector in the opposite direction as the original vector and with a new magnitude.
r R =sV@
θ +180°K θ <180°θ −180°K θ >180°
⎫ ⎬ ⎭s< 0
⎧ ⎨ ⎩
![Page 12: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/12.jpg)
Multiplying Two VectorsThere are two ways of multiplying one vector by another.
1. Scalar (dot) product…yields a scalar. r A ⋅
r B =s
r A ⋅
r B =AxBx +AyBy
r A
r B
θ
r A ⋅
r B =ABcosθ
1. The scalar product is commutative : r A ⋅
r B =
r B ⋅
r A
2. If two vectors are pependicular(normal) their scalar product is zero (0) since θ = 90° and cos 90° = 0.
θ is the smaller of the two angles
between r A and
r B .
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2. Vector (cross) product…yields a vector. r A ×
r B =
r C
r C =C=ABsinθ
The magnitude of the cross-product vector is given by:
The direction of the cross-product vector C is perpendicular to the plane generated by the vectors A and B and is given by the “right-hand-rule:
Point the fingers of the right hand in the direction of vector A so they could be curled in the direction of vector B. The thumb of the right hand will point in the direction of vector C.
θ is the smaller of the two angles
between r A and
r B .
![Page 14: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/14.jpg)
r A
r B
θ
r C =
r A ×
r B
The vector product is not commutative :r A ×
r B =−
r B ×
r A
If two vectors are parallel their vector product is zero (0) since θ = 0° and sin 0° =0.
θ
r ′ C =
r B ×
r A
r B
r A
![Page 15: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/15.jpg)
PracticeGiven the following two vectors:
Find:
r A +
r B
r A −
r B
r A ⋅
r B
r A ×
r B
r A =15@ 30°
Ax =15cos30° =13
Ay =15sin30° =7.5
r B =20@ 245°
Bx =20cos245° =−8.5
By =20sin245° =−18.1
![Page 16: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/16.jpg)
r A +
r B =
r C
Cx =Ax +Bx =13 + −8.5( )=4.5
Cy =Ay+By =7.5 + −18.1( )=−10.6
r C is in Quadrant IV
θ =360° − tan−1 10.64.5( ) = 293°
r C =11.5@293°
C = Cx2 +Cy
2 = 4.5( )2 + −10.6( )2 =11.5
θ = 360° − tan−1 Cy
Cx
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
![Page 17: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/17.jpg)
r B
245°
20
r A
30°15
r A +
r B =
r C
293°
11.5
r A +
r B =
r C
Geometrical Interpretation
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r A −
r B =
r D
Dx =Ax −Bx =13− −8.5( )=21.5
Dy =Ay−By =7.5− −18.1( )=25.6
r D is in Quadrant I
θ =tan−1 25.621.5( ) = 50°
θ = tan−1 DyDx
⎛ ⎝ ⎜
⎞ ⎠ ⎟
r D = 33.4@50°
D = D x2 +D y
2 = 21.5( )2 + 25.6( )2 =33.4
![Page 19: Vectors in Two Dimensions. VECTOR REPRESENTATION A vector represents those physical quantities such as velocity that have both a magnitude and a direction.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1ae17f8b9ab059977821/html5/thumbnails/19.jpg)
r A −
r B =
r A + −1( )
r B
−1( )
r B = 20@(245° − 180°)
−1( )r B = 20@65°
r B
20
65°
r A
30°15
r A −
r B =
r D 33.4
50°
Geometrical Interpretation
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r A ⋅
r B =s
s =AxBx +AyBy
s = 13( ) −8.5( )+ 7.5( ) −18.1( )
s =−246.25
r A
30°
15
r B
245°
20s =ABcosθ
s =−245.7
245°-30°=215°
360°-215°=145°s =(15)(20)cos(145)
What is the smallest angle between A and B?
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r A ×
r B =
r E
E =ABsinθ
E =(15)(20)sin(145)
E =172.1
r A
r B
r E points "into" the
r A
r B plane.
r E
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r A
r B
r B ×
r A =
r F
r F points "out of" the
r A
r B plane.
r F
F =ABsinθF=(15)(20)sin(145)F=172.1