Calculating the following equation with 3D-vectors

[Nemanjavuk69]

Homework description

Let there be fours vectors.

[math]\vec{A} = \begin{pmatrix} -4 \\ -11 \\ -3 \end{pmatrix} \vec{B} = \begin{pmatrix} 6 \\ 6 \\ 3 \end{pmatrix} \vec{C} = \begin{pmatrix} \sqrt{8} \\ \sqrt{4} \\ \sqrt{13} \end{pmatrix} \vec{D} = \begin{pmatrix} -10 \\ -7 \\ -7 \end{pmatrix}[/math]
The equation to be solved is as follow...
[math]\frac{1}{3}\cdot\vec{D}\cdot \vec{B} \cdot \vec{A} - \frac{1}{4}\cdot \vec{C}[/math]
Actual solution
I know for a fact, that I should end up with a new 3D-vector, and that new 3D-vector has the X, Y, Z coordinates as followed (Numbers have been rounded doing the calculations)...
[math]\vec{E} = \begin{pmatrix} 163.29 \\ 450.5 \\ 122.1 \end{pmatrix}[/math]

My solution and question.
However, when I calculate, I end up with [math]\vec{E} = \begin{pmatrix} 330.38 \\ 231.17 \\ 231.57 \\ \end{pmatrix}[/math]So my question is, where am I going wrong?


My approach
What I know so far, is that when we deal with 3D-vectors, we deal with them as with ordinary 2D-vectors.
I know that I can use a constant [math]k[/math] and simply multiply it with all the vectors coordinates like this [math]\vec{TEST} \begin{pmatrix} k\cdot x \\ k\cdot y \\ k\cdot z \\ \end{pmatrix}[/math]I also know, that you can not "multiply" two vectors as normal, therefore I have to use the dot-product (I think you might call it something else), which says I have to multiply the first vectors x-coordinate with the second vectors x-coordinate and add it to the first vectors y-coordinate and multiply it with the second vectors y-coordinate and add it to the first vectors z-coordinate and multiply it with the second vectors z-coordinate. I looks something like this
[math](\vec{A_x} \cdot \vec{B_x}) + (\vec{A_y} \cdot \vec{B_y}) + (\vec{A_z} \cdot \vec{B_z})[/math]This returns a number, AND NOT A VECTOR.So, fair enough, I use this knowledge (which I hopefully understood correctly) and apply it to the equation.
I start from left to right, since there are no parenthesis. So I say the 1/3 (constant) fraction multiplied by the vector D which gives me a new vector with the values [math]\begin{pmatrix} -3.33 \\ -2.33 \\ -2.33 \\ \end{pmatrix}[/math]I then do the dot-product between vector B and A which gives me [math]-99[/math].
I then further proceed to use that number and multiply it on my new vector, which gives me [math]\begin{pmatrix} 329.67 \\ 230.67 \\ 320.67 \end{pmatrix}[/math].

This is then where I stop, since there is no point in doing the last minus operation, since my answer even after the last operation will not even come close to the actual solution. This leads me to think, that perhaps there is an order of operation when dealing with vectors?

Ps. I am not entirely sure if this is the right board to post my question

 

[Deleted member 4993]

Homework description

Let there be fours vectors.

[math]\vec{A} = \begin{pmatrix} -4 \\ -11 \\ -3 \end{pmatrix} \vec{B} = \begin{pmatrix} 6 \\ 6 \\ 3 \end{pmatrix} \vec{C} = \begin{pmatrix} \sqrt{8} \\ \sqrt{4} \\ \sqrt{13} \end{pmatrix} \vec{D} = \begin{pmatrix} -10 \\ -7 \\ -7 \end{pmatrix}[/math]
The equation to be solved is as follow...
[math]\frac{1}{3}\cdot\vec{D}\cdot \vec{B} \cdot \vec{A} - \frac{1}{4}\cdot \vec{C}[/math]
Actual solution
I know for a fact, that I should end up with a new 3D-vector, and that new 3D-vector has the X, Y, Z coordinates as followed (Numbers have been rounded doing the calculations)...
[math]\vec{E} = \begin{pmatrix} 163.29 \\ 450.5 \\ 122.1 \end{pmatrix}[/math]

My solution and question.
However, when I calculate, I end up with [math]\vec{E} = \begin{pmatrix} 330.38 \\ 231.17 \\ 231.57 \\ \end{pmatrix}[/math]So my question is, where am I going wrong?


My approach
What I know so far, is that when we deal with 3D-vectors, we deal with them as with ordinary 2D-vectors.
I know that I can use a constant [math]k[/math] and simply multiply it with all the vectors coordinates like this [math]\vec{TEST} \begin{pmatrix} k\cdot x \\ k\cdot y \\ k\cdot z \\ \end{pmatrix}[/math]I also know, that you can not "multiply" two vectors as normal, therefore I have to use the dot-product (I think you might call it something else), which says I have to multiply the first vectors x-coordinate with the second vectors x-coordinate and add it to the first vectors y-coordinate and multiply it with the second vectors y-coordinate and add it to the first vectors z-coordinate and multiply it with the second vectors z-coordinate. I looks something like this
[math](\vec{A_x} \cdot \vec{B_x}) + (\vec{A_y} \cdot \vec{B_y}) + (\vec{A_z} \cdot \vec{B_z})[/math]This returns a number, AND NOT A VECTOR.So, fair enough, I use this knowledge (which I hopefully understood correctly) and apply it to the equation.
I start from left to right, since there are no parenthesis. So I say the 1/3 (constant) fraction multiplied by the vector D which gives me a new vector with the values [math]\begin{pmatrix} -3.33 \\ -2.33 \\ -2.33 \\ \end{pmatrix}[/math]I then do the dot-product between vector B and A which gives me [math]-99[/math].
I then further proceed to use that number and multiply it on my new vector, which gives me [math]\begin{pmatrix} 329.67 \\ 230.67 \\ 320.67 \end{pmatrix}[/math].

This is then where I stop, since there is no point in doing the last minus operation, since my answer even after the last operation will not even come close to the actual solution. This leads me to think, that perhaps there is an order of operation when dealing with vectors?

Ps. I am not entirely sure if this is the right board to post my question

I am assuming the "·" indicates multiplication. In vector field, there are two types "product", namely "dot-product " and "cross-product". Which type of product are you using ?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[LCKurtz]

When you write [imath]\frac 1 3\cdot \vec D \cdot \vec B \cdot \vec A[/imath]
you are using [imath]\cdot[/imath] for both multiplying a scalar by a vector and for a dot product of two vectors. You need to distinguish these operations. You can't dot three vectors together. Please retype it correctly or give us a copy of the original source of the problem.

 

[Nemanjavuk69]

I am assuming the "·" indicates multiplication. In vector field, there are two types "product", namely "dot-product " and "cross-product". Which type of product are you using ?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

Hello Subhotosh Khan. I have very much elaborated my approach, I even made a heading called My approach. There I am going through my problem. And regarding your question, I am using "dot product". "Cross product" is on the next chapter, so I would assume we are not meant to be using cross product yet?

 

[Nemanjavuk69]

When you write [imath]\frac 1 3\cdot \vec D \cdot \vec B \cdot \vec A[/imath]
you are using [imath]\cdot[/imath] for both multiplying a scalar by a vector and for a dot product of two vectors. You need to distinguish these operations. You can't dot three vectors together. Please retype it correctly or give us a copy of the original source of the problem.

The excercise I was given uses the "." and nothing else. Further down in my post, I have stated the difference between actually multiplying a vector and using the dot product. Please let me know if you can see it

 

[LCKurtz]

Apparently this is supposed to be some kind of puzzle to see if you can get their answer by choosing the correct interpretations for the ambiguous [imath]\cdot[/imath] operations. You seem to understand the difference between a dot product of two vectors and multiplication of a vector by a scalar. Perhaps you can change some of the dots to [imath]x[/imath] to represent scalar multiplication and leave the other dots to represent dot product and maybe insert some parentheses for clarity and find a way to get their answer. Good luck with it. This kind of problem doesn't interest me.

 

[Nemanjavuk69]

Apparently this is supposed to be some kind of puzzle to see if you can get their answer by choosing the correct interpretations for the ambiguous [imath]\cdot[/imath] operations. You seem to understand the difference between a dot product of two vectors and multiplication of a vector by a scalar. Perhaps you can change some of the dots to [imath]x[/imath] to represent scalar multiplication and leave the other dots to represent dot product and maybe insert some parentheses for clarity and find a way to get their answer. Good luck with it. This kind of problem doesn't interest me.

Hey again. I posted the equation EXACTLY as the exercise description. There are no parenthesis and the exercise ONLY uses the ".".
I was only posting this here to get some help on where/ what I did wrong, since I was not getting the correct answer. If this problem does not interest you, that's fine, no one is forcing anyone. Have a good day!

 

[Deleted member 4993]

Hey again. I posted the equation EXACTLY as the exercise description. There are no parenthesis and the exercise ONLY uses the ".".
I was only posting this here to get some help on where/ what I did wrong, since I was not getting the correct answer. If this problem does not interest you, that's fine, no one is forcing anyone. Have a good day!

I am sure you saw (response #3)

You can't dot three vectors together.

If the ACTUAL problem had three consecutive dot products of vectors - than it is not "EVEN wrong" - it is nonsense. Dot product implies - projection of one vector onto the other. You CANNOT create a projection of a vector onto a number.

That is why we want to see a photograph of the actual problem.

 

[Nemanjavuk69]

I am sure you saw (response #3)

If the ACTUAL problem had three consecutive dot products of vectors - than it is not "EVEN wrong" - it is nonsense. Dot product implies - projection of one vector onto the other. You CANNOT create a projection of a vector onto a number.

That is why we want to see a photograph of the actual problem.

First of, the picture I have is in Danish, there is no more to it than what I typed. However, since you persits, you can have the picture. It only says there are 4 vectors, A, B, C and D, and their coordinates, and that I have to calculate the equation. As you can see, they use the "." everywhere.

 

[topsquark]

It would appear that [imath]\textbf{E} = \dfrac{1}{3} ( \textbf{D} \cdot \textbf{B} ) \textbf{A} - \dfrac{1}{4} \textbf{C}[/imath] will do the trick.

-Dan

 

[Dr.Peterson]

Hey again. I posted the equation EXACTLY as the exercise description. There are no parenthesis and the exercise ONLY uses the ".".
I was only posting this here to get some help on where/ what I did wrong, since I was not getting the correct answer.

Please understand that the complaints are not against you, but against the authors of the problem. In looking at the actual problem, we can confirm that the confusion is their fault, primarily. The notation used in the problem, using the dot to mean both scalar multiplication and scalar product, whichever makes sense, is, at the least, non-standard. (For instance, here you never see a scalar multiplication written with a dot.)

But @topsquark has recognized that they appear to be reading it from left to right, taking the first dot in the actual problem (you wrote a dot also after the 1/3, which they did not) as a dot product, resulting in a scalar, and taking the next dot (non-standardly) as a scalar multiplication on vector A.

And it sounds like you intended to do that:

I start from left to right, since there are no parenthesis. So I say the 1/3 (constant) fraction multiplied by the vector D which gives me a new vector with the values

Unfortunately, you did not continue left to right, but did the dot product of B and A first. If you had done what you said, it would evidently have been correct.

The remaining question is, were you explicitly taught that the dot can be used in both ways? If so, then that is presumably part of Danish usage, or your own course. We'd benefit from knowing that; mathematics notation is not as universal as one would expect!

I'm also curious: the image shows "ewline" in many places, which seems not to have bothered you. I take it to be an error in their code, which presumably had a backslash followed by "newline", and was interpreted differently than intended. Interesting!

 

[Nemanjavuk69]

Please understand that the complaints are not against you, but against the authors of the problem. In looking at the actual problem, we can confirm that the confusion is their fault, primarily. The notation used in the problem, using the dot to mean both scalar multiplication and scalar product, whichever makes sense, is, at the least, non-standard. (For instance, here you never see a scalar multiplication written with a dot.)

But @topsquark has recognized that they appear to be reading it from left to right, taking the first dot in the actual problem (you wrote a dot also after the 1/3, which they did not) as a dot product, resulting in a scalar, and taking the next dot (non-standardly) as a scalar multiplication on vector A.

And it sounds like you intended to do that:

Unfortunately, you did not continue left to right, but did the dot product of B and A first. If you had done what you said, it would evidently have been correct.

The remaining question is, were you explicitly taught that the dot can be used in both ways? If so, then that is presumably part of Danish usage, or your own course. We'd benefit from knowing that; mathematics notation is not as universal as one would expect!

I'm also curious: the image shows "ewline" in many places, which seems not to have bothered you. I take it to be an error in their code, which presumably had a backslash followed by "newline", and was interpreted differently than intended. Interesting!

First of, my professor meant to write "newline" instead of "ewline", doing that would give a normal "matrix", like the ones on the previous excercises

Furthermore, I actually did go from left to right, since I was under the impression, that order of operation was not an issue when dealing with ".", that's why I did the subtraction operation lastly. Please quote me if it that is wrong, when dealing with vectors, that dot-products have higher priority than multiplication.

Moreover, your third question about where I was taught to be using "dot" can be used both ways. We use dot both for multiplication AND for vectors, we don't use the fancy "x" you might use in America. I only placed a dot between the fractions to show there is a multiplication involved, since, just like variables, 3x actually means 3*x.

In addition, I also checked out your link. Would this mean, when dealing with vectors, dot is only preserved to vectors meaning dot product, and there, we "remove" "." from multiplication making them implicit?

Lastly, I want to thank you so much for an elaboration on my post!

 

[Nemanjavuk69]

It would appear that [imath]\textbf{E} = \dfrac{1}{3} ( \textbf{D} \cdot \textbf{B} ) \textbf{A} - \dfrac{1}{4} \textbf{C}[/imath] will do the trick.

-Dan

Can I ask why we do [math]E= \frac{1}{3}(D⋅B)A− \frac{1}{4}C[/math] AND NOT [math]E= \frac{1}{3}(D⋅B\cdot A)− \frac{1}{4}C[/math] where we take the dot product of the first two vectors and than of the last vector?

 

[Nemanjavuk69]

It would appear that [imath]\textbf{E} = \dfrac{1}{3} ( \textbf{D} \cdot \textbf{B} ) \textbf{A} - \dfrac{1}{4} \textbf{C}[/imath] will do the trick.

-Dan

Following your instructions, I got the correct answer. So for that, thank you so much Dan!
However, I am still curious at how you knew you were supossed to do the dot product between D and B FIRST. I was under the impression that you would multiply D with 1/3.

 

[Dr.Peterson]

Can I ask why we do [math]E= \frac{1}{3}(D⋅B)A− \frac{1}{4}C[/math] AND NOT [math]E= \frac{1}{3}(D⋅B\cdot A)− \frac{1}{4}C[/math] where we take the dot product of the first two vectors and than of the last vector?

But when you take the dot product of D and B, the result is a scalar, so you can't take a dot product next! It has to be a scalar product. And that is what topsquark did.

Following your instructions, I got the correct answer. So for that, thank you so much Dan!
However, I am still curious at how you knew you were supposed to do the dot product between D and B FIRST. I was under the impression that you would multiply D with 1/3.

It doesn't really make a difference. You'll get the same result if you do as you suggest, and calculate this:

[imath]((\dfrac{1}{3} \textbf{D}) \cdot \textbf{B} ) \textbf{A} - \dfrac{1}{4} \textbf{C}[/imath]​

This is one of the properties of the dot product.

 

[Nemanjavuk69]

But when you take the dot product of D and B, the result is a scalar, so you can't take a dot product next! It has to be a scalar product. And that is what topsquark did.


It doesn't really make a difference. You'll get the same result if you do as you suggest, and calculate this:

[imath]((\dfrac{1}{3} \textbf{D}) \cdot \textbf{B} ) \textbf{A} - \dfrac{1}{4} \textbf{C}[/imath]​

This is one of the properties of the dot product.

I get it. Thank you so much for our help. I am relief to finally understand. Have a blessed week!