Whats behind the result of dot product?
Author
I know that with dot product we can obtain angles, also we can test if two vectors are in same direction, opposite direction or perpendicular. But what do numbers mean, eg. if we recieve in some dot product result 2 and in some other 12? Whats the difference between 2 and 12?
Dot product means the length, when a vector is projected to a vector of unit length.
l=(vect1,vect2), if vect2 is one unit long, l is the length, when you project vect1, to vect2.
wiki, as always
l=(vect1,vect2), if vect2 is one unit long, l is the length, when you project vect1, to vect2.
wiki, as always
the dot product derives from Pythagoras' theorem, the right angle rule. It applies when 'x' and 'y' represents values on an orthogonal system. Then the theorem applies, and the dot product equals :
a . b = |a| * |b| * cos(a, b)
if you have two dot products, one equal 2 and one equal 12, there could be many possibilities.
1) result = 2.
vector a has length 2, and is aligned with a unit vector b.
result 12.
vector a has length 12 and aligned with vector b.
2) result = 2.
both vectors have length 2, and a is at 60 degress angle from vector b.
result = 12.
a has length 24,and is at 60 degree angle from unit vector b.
ect...
You can't really find anything significant unless you know two of the three things. The length of a, the length of b, and their relative angle. Then you can derive the third unknown.
a . b = |a| * |b| * cos(a, b)
if you have two dot products, one equal 2 and one equal 12, there could be many possibilities.
1) result = 2.
vector a has length 2, and is aligned with a unit vector b.
result 12.
vector a has length 12 and aligned with vector b.
2) result = 2.
both vectors have length 2, and a is at 60 degress angle from vector b.
result = 12.
a has length 24,and is at 60 degree angle from unit vector b.
ect...
You can't really find anything significant unless you know two of the three things. The length of a, the length of b, and their relative angle. Then you can derive the third unknown.
Given two vectors, a and b, the dot product between them, a·b, is the magnitude of the projection of a onto b, multiplied by the magnitude of b.
If b is a unit vector then obviously it's magnitude is 1, so the multiplication does nothing and it's just the magnitude of the projection of a in b's direction.
If a == b then they're the same so I can collapse them into one variable for simplicity, c, the projection of c onto c is obviously c. So the result is just the magnitude of c multiplied by the magnitude of c, i.e. it's |c|2
If b is a unit vector then obviously it's magnitude is 1, so the multiplication does nothing and it's just the magnitude of the projection of a in b's direction.
If a == b then they're the same so I can collapse them into one variable for simplicity, c, the projection of c onto c is obviously c. So the result is just the magnitude of c multiplied by the magnitude of c, i.e. it's |c|2
Author
Quote:Original post by dmatter
Given two vectors, a and b, the dot product between them, a·b, is the magnitude of the projection of a onto b, multiplied by the magnitude of b.
Couldn't we than say it's the projection of b onto a multiplied by the magnitude of a. Is this the same thing? Since we have all multiplication and multiplication is commutative it should be the same thing (considering again b>a from picture).
I think there's yet one thing I could add..
This is about how the angle between the "dotted" vectors affects the result. If you understood how the dot product represents length of a vector projected onto another you might have already noticed this yourself.
Consider two vectors, a and b, both with nonzero length. Now, if the angle beetween them is (in degrees)
1) 0
cos(a, b) = 1 so the dot product is |a||b|.
2) > 0 and < 90
0 < cos(a, b) < 1 so we know that 0 < a.b < |a||b|
Most importantly, the dot product is positive
3) 90
cos(a, b) = 0 which immediately yields a.b = 0
4) > 90 and < 180
-1 < cos(a, b) < 0 so we know that -|a||b| < a.b < 0
In this case the dot product is negative
5) 180
cos(a, b) = -1 so the dot product is -|a||b|
This is about how the angle between the "dotted" vectors affects the result. If you understood how the dot product represents length of a vector projected onto another you might have already noticed this yourself.
Consider two vectors, a and b, both with nonzero length. Now, if the angle beetween them is (in degrees)
1) 0
cos(a, b) = 1 so the dot product is |a||b|.
2) > 0 and < 90
0 < cos(a, b) < 1 so we know that 0 < a.b < |a||b|
Most importantly, the dot product is positive
3) 90
cos(a, b) = 0 which immediately yields a.b = 0
4) > 90 and < 180
-1 < cos(a, b) < 0 so we know that -|a||b| < a.b < 0
In this case the dot product is negative
5) 180
cos(a, b) = -1 so the dot product is -|a||b|
The dot product (inner product) of two vectors is a measure of how similar the two vectors are (how much one is projected onto the other), multiplied by the magnitudes. Two vectors that are orthogonal don't have any amount of the other per say, so the dot product is zero. If they are parallel it's just the product of the magnitude. a.b = |a||b|cos(angle)
A matrix operating on a vector is just a series of inner products. Each row or column inner producted with the vector. Matrix multiplication is an array of inner products.
You can also inner product functions together as a way of determining 'how much' of one function is in another. For example you can take a periodic function, and pull out the different frequencies, this is the foundation of Fourier Analysis.
In Tensor Algebra you can inner product things together in any way you want.
A matrix operating on a vector is just a series of inner products. Each row or column inner producted with the vector. Matrix multiplication is an array of inner products.
You can also inner product functions together as a way of determining 'how much' of one function is in another. For example you can take a periodic function, and pull out the different frequencies, this is the foundation of Fourier Analysis.
In Tensor Algebra you can inner product things together in any way you want.
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