Скачать или смотреть Dot Products of two vectors in Physics

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Hello this is lecturer asad ali youtube channel. Here we are going to discuss First Year Physics chapter 2 vectors and Equilibrium for complete playlist of this chapter click the below links\

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hello this is asad
dear students today we will discuss
Dot product of the two vectors
we also called scalar product of two vectors
its mean that when we multiplied two vectors and we get scalar quantity.
then we recall this is dot product of two vectors.
we presented dot product by dot so that why we call its dot product .
dot product gives us scalar Quantity
for vector product we use cross sign
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Properly we define dot product as
Let A and B are non zero vectors then dot product of these vectors is define as
it is necessary for dot product both vectors are non zero
if any one vectors is zero. so cannot find dot product
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Mathematically we define as
A.B= ABcos(theta) or
A.B= |A||B|cos(theta)
where theta is the Angle between these vectors
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Graphically Interpretation
in the Graphical representation how we can prove that the Dot Product definitions
here i draw vector A and B
in the first case the magnitude of vector A is less then B.
In the second case the magnitude of vector A is greater then B.
we find Projection of B on A.
draw a perpendicular from point B to p.

here op = Projection of B on A.
here in the right angle triangle
cos(theta)= base/ hypotenuses
cos(Theta)= OP/ B
OP= B cos (theta)
Now the Dot product of the two vectors is also defined as
A.B = (Projections of B on A) * (Magnitude of the vectors)
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Similarly
we find Projection of B on A.
draw a perpendicular from point B to p.

here op = Projection of B on A.
here in the right angle triangle
cos(theta)= base/ hypotenuses
cos(Theta)= OP/ B
OP= B cos (theta)
Now the Dot product of the two vectors is also defined as
A.B = (Projections of B on A) * (Magnitude of the vectors)
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Now We discuss the angles

when theta = 0 degree
then the dot product
A.B= ABcos(0)
here cos (0)= 1
in this case vectors are parallels
this is maximum results dot Products
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when theta = 90 degree
then the dot product
A.B= ABcos(90)
here cos (90)= 0
in this case vectors are Perpendicular
this is minimum results dot Products
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when theta = 180 degree
then the dot product
A.B= ABcos(180)
here cos (180)= 0
in this case vectors are anti parallel
this is negative results results dot Products
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Unit Vectors
i.i=1
j.j=1
k.k=1
by using definitions
i.i= |i||i|cos(0)= 1
similarly

j.j= |j||j|cos(0)= 1

i.j= |i||j|Bcos(90)= 0