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8/13/2019 2Motion in Which Direction_Vectors & Scalars
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Topic 2
Motion in which
direction?
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Motionin which direction?
Motion in which direction?
Representing vector quantities
Graphical addition of vectors Subtracting vectors
Relative velocities: Some applications
Components of vectorAlgebraic addition of vectors
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Vectors and Scalars
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Vectors and Scalar
Scalar quantityis described by a single realnumber, or magnitude, it can be positive, zero ornegative
Vector quantityhas both a magnitude and adirection in space.
Vectors are represented by a bold face type(e.g.,A).
Alternatively, we can also write it as A
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Vectors: geometric representation
Direction of the arrow gives the direction of the vector
Length of the arrow gives the magnitude of the vector
Head
Tail
A vector is geometrically represented as a line segment
with an arrow indicating direction
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Displacement
Displacementis the change of position of a
point
P2
P1
A
The displacement formpoint P1 to P2is vector
A
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Displacement
Displacement is always a
straight-line segment
directed from the starting
point to the end point If the path ends at the
same place where it
started, the displacement
is zero
P2
P1 Displacement is not relateddirectlyto the distance traveled
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Vector Addition
A
B
C =A+B
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A B
C
D
R
R = ( A+B ) + C
= D + C
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Addition of vectors
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Components of Vectors
Definition of Vector Components
In two dimensions, the vector components of avectorAare two perpendicular vectorsAxandAythat are parallel to the xandyaxes, respectively,
and add together vectorially so that
A = Ax+ Ay
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Components of Vectors
Ax
Ay Acos
sin
tan
x y
x
y
y
x
A A A
A
AA
A
A
A
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Resolving a Vector into components
xA
yA
x
yA
x yA A cos , A A sin
2 2
x yA A A
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R
A
B
Ax Bx
Rx
By
Ay
Ry
2 2
x x x
y x y
x y
R A B
R A B
A A A
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A unit vector ris a vector having length 1 and no units.
It is used to specify the direction of a vector:
So we can writeA= Ar
The unit vectors i, j, k point in the x,yand z axesrespectively.
x
y
zi
j
k
A
r^
In terms of unit vectors, we can express a 2-D
vector as follows:
x yA A A i j
UNIT VECTORS
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Summary
Scalar quantities are numbers, and combine with the usual rulesof arithmetic.
Vector quantities have direction as well as magnitude, andcombine according to rules of vector addition.
Graphically, two vectorsAand Bare added by placing the tail ofBat the head, or tip, ofA.
The vector sumA+ Bthen extends from the tail ofAto thehead of B.
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An airplane flies with a velocity of 750
kilometers per hour, 30.0south of east. What is
the magnitude of Vx and Vy ?
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End