Scalar divided by a Vector

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TheFarmerLad
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#1
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#1
How do you simplify a scalar divided by a vector?

For example if we have

an unknown vector (say x y z) = a known scalar (say lambda) / a known vector (say i j k), how do you find out the values of x, y and z in terms of lamda, i, j and k?
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Tayls102
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#2
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Do you have a specific question that you could perhaps screenshot/ share a picture of?

Straight division/ multiplication by vectors aren't defined operations.

There are two main operations on vectors which can be compared to the ordinary multiplication of numbers but are slightly different for vectors, namely the dot product and the cross product.

We also have normal addition of vectors that work as you would expect, i.e add component wise, e.g (1,2,3)+(4,5,6)=(5,7,9).
And we can multiply a vector by a scalar, so for example 2*(1,2,3)=(2,4,6).

So something like (lambda)/(a vector) doesn't really make any sense, or at least isn't defined mathemtically. Unless there is an function applied to the vector in the denominator, for example the norm function which takes a vector (x,y,z) and returns the magnitude of the vector sqrt(x2+y2+z2).
Last edited by Tayls102; 2 years ago
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TheFarmerLad
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(Original post by Tayls102)
Do you have a specific question that you could perhaps screenshot/ share a picture of?

Straight division/ multiplication by vectors aren't defined operations.

There are two main operations on vectors which can be compared to the ordinary multiplication of numbers but are slightly different for vectors, namely the dot product and the cross product.

We also have normal addition of vectors that work as you would expect, i.e add component wise, e.g (1,2,3)+(4,5,6)=(5,7,9).
And we can multiply a vector by a scalar, so for example 2*(1,2,3)=(2,4,6).

So something like (lambda)/(a vector) doesn't really make any sense, or at least isn't defined mathemtically. Unless there is an function applied to the vector in the denominator, for example the norm function which takes a vector (x,y,z) and returns the magnitude of the vector sqrt(x2+y2+z2).
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Tayls102
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#4
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Okay so I have a feeling that here w3 is a vector rather than a scalar. There are two reasons why I think this, the first is that the use of 'x' in the context of vectors really means cross product, which is only defined on vectors (i.e you can't take the cross product between a scalar and a vector). The second reason is that if you do treat it as a scalar, and interpret 'x' as normal multiplication by a scalar, what you get doesn't really make sense. Of course, I can't be sure without some context of the question.

Does this help?
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TheFarmerLad
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(Original post by Tayls102)
Okay so I have a feeling that here w3 is a vector rather than a scalar. There are two reasons why I think this, the first is that the use of 'x' in the context of vectors really means cross product, which is only defined on vectors (i.e you can't take the cross product between a scalar and a vector). The second reason is that if you do treat it as a scalar, and interpret 'x' as normal multiplication by a scalar, what you get doesn't really make sense. Of course, I can't be sure without some context of the question.

Does this help?
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Tayls102
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Okay, lets assume that w3 is a vector which I'm fairly sure it would be.

Are you familiar with the cross product of two vectors?
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TheFarmerLad
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#7
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(Original post by Tayls102)
Okay, lets assume that w3 is a vector which I'm fairly sure it would be.

Are you familiar with the cross product of two vectors?
yes. to my knowledge the above will produce three simulataneous equations, I just don't know how to solve them
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Tayls102
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#8
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Great! that sounds right, what system of equations* did you get?

What methods do you know to solve a system of equations?

Edit: *

Spoiler:
Show
Hint: There may or may not be a unique solution
Last edited by Tayls102; 2 years ago
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TheFarmerLad
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#9
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(Original post by Tayls102)
Great! that sounds right, what system of equations* did you get?

What methods do you know to solve a system of equations?

Edit: *
Spoiler:
Show
Hint: There may or may not be a unique solution.
i get this but have no idea how to find wx, wy and wz
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Tayls102
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#10
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That looks good, is there perhaps a way you could write this in the form Aw3=v, where v is a vector, and A is a 3x3 matrix, both to be found?

Also, remember that wx,wy,wz are components of w3 so don't need to be in bold
Last edited by Tayls102; 2 years ago
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