Algebraic Manipulation of Vector Equations (Linear Algebra To Algebra Analogy)

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programmerBlack

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Hello,

I've been studying Linear Algebra, and have been working with Vectors. I'm pretty familiar with many of the vector concepts, Normalization, Vector addition and subtraction, Dot Products and the like.

The question I have is under what subject does the "Algebraic Manipulation" of vector equations fall under? I cannot find any resources on the subject. I find myself having a hard time manipulating equations that feature operations like the Dot Product. What I"m looking for is, what are the rules that define the "What you do to one side, you should do to the other side" concepts when vectors are involved in an Algebraic equation?

Take for example, the Dot Product definition: a * b = |a| * |b| * cos(theta) where a, and b are vectors.

If I was given then vector a, and theta, how would I go about solving for b?

This obviously would require some algebraic manipulations. And at some point I would be required to divide both sides by the vector a. Which immediately doesn't make sense to me. And there lies my problem. Where can I find out how to manipulate this equation "algebraically", and still be able to work with vectors as inputs.

Thanks in advance for any responses. Much appreciated.
 
programmerBlack said:
Take for example, the Dot Product definition: a * b = |a| * |b| * cos(theta) where a, and b are vectors.

If I was given then vector a, and theta, how would I go about solving for b?
Click to expand...
Is this an actual problem you had to deal with?
 
programmerBlack said:
Hello,

I've been studying Linear Algebra, and have been working with Vectors. I'm pretty familiar with many of the vector concepts, Normalization, Vector addition and subtraction, Dot Products and the like.

The question I have is under what subject does the "Algebraic Manipulation" of vector equations fall under? I cannot find any resources on the subject. I find myself having a hard time manipulating equations that feature operations like the Dot Product. What I"m looking for is, what are the rules that define the "What you do to one side, you should do to the other side" concepts when vectors are involved in an Algebraic equation?

Take for example, the Dot Product definition: a * b = |a| * |b| * cos(theta) where a, and b are vectors.

If I was given then vector a, and theta, how would I go about solving for b?

This obviously would require some algebraic manipulations. And at some point I would be required to divide both sides by the vector a. Which immediately doesn't make sense to me. And there lies my problem. Where can I find out how to manipulate this equation "algebraically", and still be able to work with vectors as inputs.

Thanks in advance for any responses. Much appreciated.
Click to expand...
The part a.b has to be obtained by some other method (i.e. without using the given equation) - then you could divide that by magnitude of vector a (= |a|) and cos(theta) to get magnitude of vector b (= |b|).

By the way, dividing by vector is not defined (however you can find "product" of two vectors in different ways).
 
programmerBlack said:
I've been studying Linear Algebra, and have been working with Vectors. I'm pretty familiar with many of the vector concepts, Normalization, Vector addition and subtraction, Dot Products and the like.
The question I have is under what subject does the "Algebraic Manipulation" of vector equations fall under? I cannot find any resources on the subject. I find myself having a hard time manipulating equations that feature operations like the Dot Product. What I"m looking for is, what are the rules that define the "What you do to one side, you should do to the other side" concepts when vectors are involved in an Algebraic equation?
Take for example, the Dot Product definition: a * b = |a| * |b| * cos(theta) where a, and b are vectors.
If I was given then vector a, and theta, how would I go about solving for b?
Click to expand...
Please post an actual problem that you would like help with.
That will give us some idea what you mean by the above.
 
Subhotosh Khan said:
The part a.b has to be obtained by some other method (i.e. without using the given equation) - then you could divide that by magnitude of vector a (= |a|) and cos(theta) to get magnitude of vector b (= |b|).

By the way, dividing by vector is not defined (however you can fin "product" of two vectors in different ways).
Click to expand...

Thanks, this is very close to an answer I was looking for. In response to Pka, I'm currently working on some code, and I need to find a vector given, say the Dot Product equation above. Let's say I defined it as: a * b = |a| * |b| * cos(theta) (Same as above), and I was given 2 variables of the 3.

That would bring the equation to look something like this: a_x * 1 + a_y * 0 + a_z * 0 = |a| * |(1, 0, 0)| * cos(90 degrees).
The question I'm asking is how would I find the vector a = (a_x, a_y, a_z), given vector b, and theta = 90 degrees.

I'm basically looking for a vector that when dotted with b, there 90 degrees in between them - algebraically.

Thanks again.
 
a_x * 1 + a_y * 0 + a_z * 0 = 0
a_x = 0.
Any vector with a_x = 0 would work.
Just make a drawing - what vectors are normal to the x axis? Well, any vector with 0 x coordinate.
 
lev888 said:
a_x * 1 + a_y * 0 + a_z * 0 = 0
a_x = 0.
Any vector with a_x = 0 would work.
Just make a drawing - what vectors are normal to the x axis? Well, any vector with 0 x coordinate.
Click to expand...

Got it thanks, so I also need to choose values for a_y, and a_z components. Quite literally there need to be only 1 unknown. I'm now assuming when I'm working with these kinds of equations, I should expand out the vectors to component form?
 
programmerBlack said:
Got it thanks, so I also need to choose values for a_y, and a_z components. Quite literally there need to be only 1 unknown. I'm now assuming when I'm working with these kinds of equations, I should expand out the vectors to component form?
Click to expand...
Please add some context. Where does "there need to be only 1 unknown" come from?
Regarding component form - not sure, depends. If you know that some of the coordinates are zeros, expanding may simplify things.
 
programmerBlack said:
I'm basically looking for a vector that when dotted with b, there 90 degrees in between them - algebraically.
Click to expand...
Suppose that \(\vec{b}=<1,-1,2>\) If you pick any vector \(\vec{a}\) such that \(\vec{a}\cdot\vec{b}=0\) then the two vectors are perpendicular hence the angle between them is \(\dfrac{\pi}{2}\).
Any of these willl work: \(<-2,2,2>,~<-2,0,1>,~<4,4,0>,~<x,x,0>x\ne 0\)
 
programmerBlack said:
Let's say I defined it as: a * b = |a| * |b| * cos(theta) (Same as above), and I was given 2 variables of the 3.

That would bring the equation to look something like this: a_x * 1 + a_y * 0 + a_z * 0 = |a| * |(1, 0, 0)| * cos(90 degrees).
The question I'm asking is how would I find the vector a = (a_x, a_y, a_z), given vector b, and theta = 90 degrees.

I'm basically looking for a vector that when dotted with b, there 90 degrees in between them - algebraically.
Click to expand...
The way you are asking this is extremely confusing, due to notation and other issues.

You are starting with the definition of the dot product: [MATH]\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}|\,|\mathbf{b}|\cos(\theta)[/MATH]. The two sides not only are equal in some particular case you are solving for, but mean the same thing. The equation is true for any two vectors; but theta is defined as the angle between the two vectors, and is not independent of them.

It appears that you are saying that you would be given a vector b (in three-dimensional space) and an angle theta, and want to find a vector a such that the angle between them is theta. Do you see, just by visualizing it, that there will be infinitely many such vectors a? They will form a cone around b, so to speak. In your specific example, any vector orthogonal (perpendicular) to b will work, and those will form a plane. One way to find a vector like that would be to choose any other vector and take the cross product with b.

If you want to pose a solvable problem for us to help you with, you will need to give more information -- say, the angles between vector a and two given vectors b and c. Even then, any vector (of any length) in one of two different directions would be a solution.

Perhaps you need to give us a specific scenario that you are trying to program, rather than trying to make up a simplified example.
 
Dr.Peterson said:
The way you are asking this is extremely confusing, due to notation and other issues.

You are starting with the definition of the dot product: [MATH]\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}|\,|\mathbf{b}|\cos(\theta)[/MATH]. The two sides not only are equal in some particular case you are solving for, but mean the same thing. The equation is true for any two vectors; but theta is defined as the angle between the two vectors, and is not independent of them.

It appears that you are saying that you would be given a vector b (in three-dimensional space) and an angle theta, and want to find a vector a such that the angle between them is theta. Do you see, just by visualizing it, that there will be infinitely many such vectors a? They will form a cone around b, so to speak. In your specific example, any vector orthogonal (perpendicular) to b will work, and those will form a plane. One way to find a vector like that would be to choose any other vector and take the cross product with b.

If you want to pose a solvable problem for us to help you with, you will need to give more information -- say, the angles between vector a and two given vectors b and c. Even then, any vector (of any length) in one of two different directions would be a solution.

Perhaps you need to give us a specific scenario that you are trying to program, rather than trying to make up a simplified example.
Click to expand...

Hmmm, thanks Dr.Peterson and others. I will spend some time trying to think of the proper question to ask. Originally I was just trying to find some resources that give the rules for dealing with vectors "when used in equations". Similar to the algebra rules defined here: Algebra Cheat Sheet. Not necessarily the answer to a specific question.

I'm looking for, the opposite of Multiplication is Divison. The opposite of Addition is Subtraction. So does the Dot product have an opposite
operation? What about other Linear Algebra operations? Adding and subtracting vectors seems to work and are inverses of each other.

This is less about a specific Math Problem I was working on, and more about the operations (rules) that can be taken on equations involving vectors.

Thanks.
 
programmerBlack said:
I'm looking for, the opposite of Multiplication is Divison. The opposite of Addition is Subtraction. So does the Dot product have an opposite
operation? What about other Linear Algebra operations? Adding and subtracting vectors seems to work and are inverses of each other.
This is less about a specific Math Problem I was working on, and more about the operations (rules) that can be taken on equations involving vectors.
Click to expand...
To think of the operation of dot products in the same context as addition or multiplication is what logicians call a category error.
Both addition & multiplication are defined an a well understood set and each has an identity element, zero for addition and one for multiplication. Those two elements make what you as calling opposite operations possible. There is no corresponding notion associated with the dot product of vectors. The dot product operation assigns to two vectors a real number.
If each of \(\bf{\vec{{a}}}~\&~\bf{\vec{{b}}}\) is a vector then \(\dfrac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|~\|\vec{b}\|}=\cos(\theta)\) where \(\theta\) is the angle between the vectors.
There is just no concept of an inverse of that operation.
 
programmerBlack said:
Hmmm, thanks Dr.Peterson and others. I will spend some time trying to think of the proper question to ask. Originally I was just trying to find some resources that give the rules for dealing with vectors "when used in equations". Similar to the algebra rules defined here: Algebra Cheat Sheet. Not necessarily the answer to a specific question.

I'm looking for, the opposite of Multiplication is Divison. The opposite of Addition is Subtraction. So does the Dot product have an opposite
operation? What about other Linear Algebra operations? Adding and subtracting vectors seems to work and are inverses of each other.

This is less about a specific Math Problem I was working on, and more about the operations (rules) that can be taken on equations involving vectors.

Thanks.
Click to expand...

For a start, any textbook on linear algebra (or a book at a more general level that has an adequate section on vectors) ought to list properties, just as an ordinary algebra text will state rules for algebra with numbers. Most of it is the same: you can add the same thing to both sides of an equation, you can commute terms being added, and so on. For example,


One big difference, as has been mentioned already, is that there is no division; that is, neither the dot product nor the cross product has an inverse (though multiplication of a vector by a scalar does).

For example, an equation you might want to solve could be [MATH]\mathbf{a}\cdot\mathbf{x} = c[/MATH], where you know vector a and scalar c, but not vector x. Since this is equivalent to [MATH]|\mathbf{a}|\,|\mathbf{x}|\cos(\theta) = c[/MATH], it might be solved by dividing in that form, [MATH]|\mathbf{x}| \cos(\theta) = c\div|\mathbf{a}|[/MATH], but there both both factors on the left could be varied freely, so there is not a single solution. (x could be any vector whose projection on b has a certain value.)

On the other hand, an equation like ax + b = c can be solved the way you would solve anything over real numbers.

You could also rewrite your vector equation in terms of components; this would give you the equation of a line, any point on which would be a solution.
 
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