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Lecture 3 PHY 107)
SPRING 2016
INSTRUCTOR : SUBIR GHOSH, PHD
http://www.northsouth.edu/index.html
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Vectors
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A scalar quantity is specified by a single value with an appropriate unit and has no directiontemperature, mass, energy etc. )
A vector quantity has both magnitude and direction. (example displacement, velocity)
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Properties of Vectors
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1. Equality of Vectors:
Two vectors are equal if they have the same magnitude and point in the same
direction.
2. Adding Vectors:
The resultant vector is the vector that connects from the tail of a vector to the tip of another
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Vector Addition
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1. Commutative Law of Addition:
A+B = B+A
2. Associative Law of Addit
A+(B+C) = (A+B) +C
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Negative of a Vector
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The negative of a vector A is defined as the vector that when added to A gives zero for the v
A + (-A) = 0
The vectors A and –A have the same magnitude but point in opposite directions.
Subtracting Vectors:
A – B = A + (-B)
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Sample Problem
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A car travels 20.0km due north and then 35.0 km due west. Find the magnitude and directiresultant displacement.
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Components of a Vector
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•A component of a vector is the projection of the vector on an axis.
• The projection of a vector on x-axis is its x component and the projection
on y-axis is its y component.
• The process of finding the components of a vector is called resolving a
vector.
• Once a vector has been resolved into its components along a set of axes,the components themselves can be used in place of vector.
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Sample Problem
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A small airplane leaves an airport on an overcast day and is later sighted 215km away in a di
making an angle of 22 degree east of north. How far east and north is the airplane from the
sighted?
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Unit Vector
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• A unit vector is a vector that has a magnitude of exactly one and points in a particular dir
• It lacks both dimension and unit.
• A unit vector is denoted by a lower case letter with a hat.
• Its sole purpose is to point or specify a direction.
= +
= +
Vector Compon
ax, ay are scalar components.
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Adding Vectors by Components
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= + The vector is the same as the v
Each component of must be the same of the corresponding components of ( + ):
= +
= +
= +
Two vectors must be equal if their corresponding components are equal.
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Sample Problem
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Find the vector sum of the following three vectors:
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Sample Problem
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Two vectors must be equal if their corresponding components are equal.
Find the sum of two vectors A and B lying in the xy plane and given by
= 2.0 + 2.0 and B= 2.0 − 4.0
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Multiplying Vectors
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Multiplying a vector by a scalar:
If we multiply a vector by a scalar s, we get a new vector. Its magnitude is the product of m and the absolute value of s. Its direction is the direction of if s is positive but the opposs in negative.
Multiplying a vector by a vector:
(a) Scalar Product ( produces a scalar quantity)
(b) Vector Product (produces a vector quantity)
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Multiplying Vectors
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Scalar Product
. = ∅#1
#2
a is the magnitude of vector and b is the magnitude of vector ;∅ is the angle between the vectors
. = + +
. = ( + + ) ∙( + + )
. = .
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Sample Problem page 49)
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What is the angle φ between = 3.0 − 4.0 and = −2.0 +3 .0?
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Multiplying Vectors
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Vector Product (Also known as cross product)
The vector product of and , written as × , produces a third vector who
=
Where is the smaller of the two angles between of and .
×=−(×)
#1
#2 × = ( + + )×( + + )
× = − + − + ( −
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Multiplying Vectors
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If = 3.0 − 4.0 and = −2.0 +3 .0
, what is = × ?