Decide a matrix for a vector that goes through various morphs

In summary, the conversation discusses the correct matrix for rotation and the incorrect matrix for mirroring. The person seeking help is unsure of how to properly reflect along the x, y, and z planes and asks for an example. The expert suggests trying different options and explains that the rotation direction does not determine the reflection. The expert then provides an example of a reflection along the xy plane and confirms that it is correct.
  • #1
Wi_N
119
8
Homework Statement
1. first a quarter rotation on the x-axis. 2. then being mirrored on xy-plane. 3. Then projected on the plane x+ 2y+ 3z= 0.
Relevant Equations
$$\begin{pmatrix}
1 & 0 & 0 \\
0 & cos(\theta) & -sin(\theta) \\
0& sin(\theta) & cos(\theta)
\end{pmatrix}$$

for rotating on the x-axis.
vector=(abc)

1.
$$\begin{pmatrix}
1 & 0 & 0 \\
0 & cos(\theta) & -sin(\theta) \\
0& sin(\theta) & cos(\theta)
\end{pmatrix}$$
The rotation part is correct.

2.
$$\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0& 0 & 0
\end{pmatrix}$$ is wrong apparently

how do I do the mirroring?

step 3 i can do just fine.
 
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  • #2
Different authors use different setups for the XYZ setup. Do you know which one the author uses? Your matrix sends ##(x,y,z)## to ##(x, y,0)## which is not a reflection but a projection. In 2D, if you reflect across the y axis, you send ##(x,y)## to ##(-x,y)##. But the reflection you have will depend on how ##XYZ## axes are setup.
 
  • #3
WWGD said:
Different authors use different setups for the XYZ setup. Do you know which one the author uses? Your matrix sends ##(x,y,z)## to ##(x, y,0)## which is not a reflection but a projection. In 2D, if you reflect across the y axis, you send ##(x,y)## to ##(-x,y)##. But the reflection you have will depend on how ##XYZ## axes are setup.
it doesn't specify just that it mirrors.
 
  • #4
Wi_N said:
it doesn't specify just that it mirrors.
Then it seems you just have to try different options.
 
  • #5
WWGD said:
Then it seems you just have to try different options.

could you give one example? it said the rotation was in positive direction, does that help?
 
  • #6
Can you see what happens when you reflect a generic (x,y,z) along either of the 3 planes? This should help you define the reflection matrix. Does it help?
 
  • #7
WWGD said:
Can you see what happens when you reflect a generic (x,y,z) along either of the 3 planes? This should help you define the reflection matrix. Does it help?
not really.
 
  • #8
Ok, let's see if this works:
Draw a horizontal line L . At any point of L draw a perpendicular line segment S to L so that the endpoints of S are both equidistant to L. The endpoints of S are a point and its reflection about a plane. Can you tell the relation between the two emdpoints of S ( in coordinates)?

If not, maybe @BvU can help illustrate it better?
 
Last edited:
  • #9
ok so a reflecton in xy plane is given by

100
010
00-1
 
  • #10
is this correct?
 
  • #11
Yes, that is correct.
 
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Related to Decide a matrix for a vector that goes through various morphs

1. What is a matrix and how is it related to vectors?

A matrix is a rectangular array of numbers or symbols that are arranged in rows and columns. It is related to vectors because it can be used to represent and manipulate multiple vectors at once.

2. What are morphs in relation to matrices and vectors?

Morphs refer to the different transformations or changes that can be applied to a vector using a matrix. These transformations can include scaling, rotating, and shearing the vector.

3. How do you decide on a matrix for a vector that goes through various morphs?

The matrix is determined by the desired transformations for the vector. Each transformation can be represented by a specific matrix, and by multiplying these matrices together in a specific order, the resulting matrix can be used to achieve the desired morphs for the vector.

4. Can a single matrix achieve multiple morphs for a vector?

Yes, a single matrix can be used to achieve multiple morphs for a vector. This is because the order of multiplication of matrices can be changed, resulting in different combinations of transformations.

5. Are there any limitations to using matrices for vector morphs?

While matrices can be used to achieve a variety of transformations for vectors, there are some limitations. For example, matrices cannot be used to manipulate vectors in 3-dimensional space, as they can only represent transformations in 2-dimensional space.

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