RAI SAHEB BHANWAR SINGH COLLEGE NASRULLAGANJ

1. Vector Product of Two Vectors
BY ARADHANA SOLONKI
SUBMITTED TO GYANRAO DHOTE

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What is a Vector?
 A quantity that has both
 Size
 Direction
 Examples
 Wind
 Boat or aircraft travel
 Forces in physics
 Geometrically
 A directed line segment
Initial point
Terminal
point

3.  A vector can be multiplied by a scalar.
The components of the vector are
multiplied by the scalar and the result is
a scaled vector which in the same
direction as the original vector if the
scalar is positive, or in the opposite
direction if the scalar is negative.
 A vector can also be multiplied by
another vector. Two types of vector
multiplications have been defined,
the scalar product and the vector
product.
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4.  The scalar product or dot
product A×B of two
vectors A and B is not a vector, but a
scalar quantity (a number with units).
In terms of the Cartesian components
of the vectors A and B the scalar
product is written as
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5.  A×B = AxBx + AyBy + AzBz.
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Dot Product
Given vectors V = <a, b>, W = <c, d>
 Dot product defined as
 Note that the result is a scalar
 Also known as
 Inner product or
 Scalar product
V W a c b d= × + ×g

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Find the Dot (product)
 Given A = 3i + 7j, B = -2i + 4j, and
C = 6i - 5j
 Find the following:
 A • B = ?
 B • C = ?
 The dot product can also be found
with the following formula
cosV W V W α= × ×g

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Dot Product Formula
 Formula on previous slide may be
more useful for finding the angle α
cos
cos
V W V W
V W
V W
α
α
= × ×
=
×
g
g

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Find the Angle
 Given two vectors
 V = <1, -5> and W = <-2, 3>
 Find the angle between them
 Calculate dot product
 Then magnitude
 Then apply
formula
 Take arccos
V
W

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Dot Product Properties (pg 321)
 Commutative
 Distributive over addition
 Scalar multiplication same over dot
product before or after dot product
multiplication
 Dot product of vector with itself
 Multiplicative property of zero
 Dot products of
 i • i =1
 j • j = 1
 i • j = 0

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Assignment B
 Lesson 4.3B
 Page 325
 Exercises 37 – 61 odd

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Scalar Projection
 Given two vectors v and w
 Projwv =
v
w
projwv
The projection of v on w
α
cosv α×

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Scalar Projection
 The other possible configuration for
the projection
 Formula used is the same but result
will be negative because α > 90°
v
w projwv
The projection of v on w
α
cosv α×

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Parallel and Perpendicular Vectors
 Recall formula
 What would it mean if this resulted in a
value of 0??
 What angle has a cosine of 0?
cos
V W
V W
α
•
=
×
0 90
V W
V W
α
•
= ⇔ =
×
o

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Work: An Application of the Dot
Product
 The horse pulls for 1000ft with a force
of 250 lbs at an angle of 37° with the
ground. The amount of work done is
force times displacement. This can be
given with the dot product
37°
W F s= ×
cosF s α= × ×

19. The vector product or "cross
product"
 The vector product or "cross
product" of two vectors A and B is a
vector C, defined as C=A´B.
 We can find the Cartesian components
of C=A´B in terms of the components
of A and B.
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21.  Again, consider two arbitrary
vectors A and B and choose the
orientation of your Cartesian
coordinate system such that A points
into the x-direction and B lies in the
x-y plane. Then A = (Ax, 0, 0)
and B = (Bx, By, 0) and
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22. The magnitude of C is
C=Cz=AxBy.
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23.  Since Ax=A and By= Bsinf we can also write
 C=ABsinf,
 where f is the smallest angle between the directions of the
vectors A and B. C is perpendicular to both A and B, i.e. it is
perpendicular to the plane that contains both Aand B. The
direction of C can be found by inspecting its components or
by using the right-hand rule.
 Let the fingers of your right hand point in the direction of A.
Orient the palm of your hand so that, as you curl your fingers,
you can sweep them over to point in the direction of B. Your
thumb points in the direction of C=A´B.

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29.  If A and B are parallel or anti-parallel to each other,
then C=A´B=0, since sinf=0. If A and B are
perpendicular to each other, then sinf=1 and C has its
maximum possible magnitude.
 When we form the scalar product of two vectors, we
multiply the perpendicular component of the two
vectors. The vector product is not commutative,
 A´B = -B´A.
 Many physical quantities of interest are calculated by
forming the vector product of two vectors.
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