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APPLICATION OF CROSS PRODUCT AND DOT PRODUCT IN REAL LIFE
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UNDERSTAND INTUITIVELY.....
1] How sin( ) is a measure of how much perpendicular two objects/forces are.
2] How cos( ) is a measure of how much parallel two objects/forces are.
3] How in complex numbers
i = rotation by 90 degrees
i^2= rotation by 180 degrees
i^3= rotation by 270 degrees.
i^4 = rotation by 360 degrees.
4] Transpose of a matrix = Rotation by 180 degrees around the diagonal.
5] e^i(angle) = rotation by that angle.
6] sin(30) = 1/2
because at 30 degree the effect of a force reduces to half over the other force/object (as against when the two forces/object are against at 90 degrees w.r.t each other)
7) A X B = |A| |B| sin(angle between A and B).
Application of cross product and dot product in real life.
PDF Book Link
Blog Link
VISUALIZING MATH 2 (WORD & PDF BOOK)
PDF LINK FOR VISUALIZING MATH 2
Visualizing Math 2 deals with concepts like
1) How Fourier Transforms are the side-view of a wave.
2) Laplace Transforms are side-view + front view (Imaginary + real part)
3) How Z Transforms are nothing but the discrete cousin of Laplace and DFT of Fourier.
4) How the number 1 appears actually as a fraction 1/1 in Math which means full of full.
Thus sin(90) = 1/1 = full on influence at 90 deg
Thus Max probability = 1/1 = full of full sample space
Thus cos(0) = 1/1 = full on full influence at 0 deg (when parallel)
Unit Circle radius = 1/1 = magnitude doesn't decrease/increase
throughout the rotation. Else it would have been a ratio like 2/1 or 1/4 etc.
5) How Continous functions can be visualized as those functions which you can draw on a paper without lifting up the pencil.
6) How a function differentiable at a point means if you stand at that point,...
(1) You will experience a slope. Slope will exist.
(2) This slope however will not be infinite. (fully vertical)
(3) The slope will be either towards the left or the right but not both the directions.
(4) The slope will not wriggle like a snake (oscillate ) at that point.
7) How a hermitian matrix can be visualized as a any object which which exhitibits vertical symmetry. So even if it is rotated upside down about the central axis( diagonal of the matrix) you wont be able to tell the difference.
8) How probability can be visualized much more intuitively by multiplying it by 100 and thus converting it into a percentage.
Eg:- a probability of 0.2 can be converted to 0.2 x 100 = 20%
9) How probability density can be understood using an example of the Probability of finding a man after entering New-york v/s finding a man after entering a sparsely populated desert.
And so on.........
The book is still in works and subsequent updated will be sent to your mail id for free.
Binnoy
A SEQUEL TO VISUALIZING MATH 1
The Book Visualizing Math 2 in word and pdf format is a sequel to Visualizing Math which is once again available on gumroad the link as
Visualizing Math 1 Link
1] How sin( ) is a measure of how much perpendicular two objects/forces are.
2] How cos( ) is a measure of how much parallel two objects/forces are.
3] How in complex numbers
i = rotation by 90 degrees
i^2= rotation by 180 degrees
i^3= rotation by 270 degrees.
i^4 = rotation by 360 degrees.
4] Transpose of a matrix = Rotation by 180 degrees around the diagonal.
5] e^i(angle) = rotation by that angle.
6] sin(30) = 1/2
because at 30 degree the effect of a force reduces to half over the other force/object (as against when the two forces/object are against at 90 degrees w.r.t each other)
7) A X B = |A| |B| sin(angle between A and B).
Application of cross product and dot product in real life.
PDF Book Link
Blog Link
VISUALIZING MATH 2 (WORD & PDF BOOK)
PDF LINK FOR VISUALIZING MATH 2
Visualizing Math 2 deals with concepts like
1) How Fourier Transforms are the side-view of a wave.
2) Laplace Transforms are side-view + front view (Imaginary + real part)
3) How Z Transforms are nothing but the discrete cousin of Laplace and DFT of Fourier.
4) How the number 1 appears actually as a fraction 1/1 in Math which means full of full.
Thus sin(90) = 1/1 = full on influence at 90 deg
Thus Max probability = 1/1 = full of full sample space
Thus cos(0) = 1/1 = full on full influence at 0 deg (when parallel)
Unit Circle radius = 1/1 = magnitude doesn't decrease/increase
throughout the rotation. Else it would have been a ratio like 2/1 or 1/4 etc.
5) How Continous functions can be visualized as those functions which you can draw on a paper without lifting up the pencil.
6) How a function differentiable at a point means if you stand at that point,...
(1) You will experience a slope. Slope will exist.
(2) This slope however will not be infinite. (fully vertical)
(3) The slope will be either towards the left or the right but not both the directions.
(4) The slope will not wriggle like a snake (oscillate ) at that point.
7) How a hermitian matrix can be visualized as a any object which which exhitibits vertical symmetry. So even if it is rotated upside down about the central axis( diagonal of the matrix) you wont be able to tell the difference.
8) How probability can be visualized much more intuitively by multiplying it by 100 and thus converting it into a percentage.
Eg:- a probability of 0.2 can be converted to 0.2 x 100 = 20%
9) How probability density can be understood using an example of the Probability of finding a man after entering New-york v/s finding a man after entering a sparsely populated desert.
And so on.........
The book is still in works and subsequent updated will be sent to your mail id for free.
Binnoy
A SEQUEL TO VISUALIZING MATH 1
The Book Visualizing Math 2 in word and pdf format is a sequel to Visualizing Math which is once again available on gumroad the link as
Visualizing Math 1 Link
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