Then point $(3, 3)$ transformed into what point?
I couldn't solve this problem, I need step by step solution to understand and learn. Does it matter two dimensional or three dimensional? What if rotation angle other than $45^\circ$.?
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So did I understand it correctly, that in the beginning we have a Cartesian coordinate system. A new coordinate system is made by turning the old coordinates $45^{\circ}$ (anti-clockwise?) about the point $(2,1)$. What are the new coordinates of a point $A$, which was represented by the point $(3,3)$ in the old system? – Matti P. Jun 28 '19 at 08:09
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In any case, I would approach this by first "moving" the coordinate sytem so that the rotation is carried out around the origin. This is because the rotation transformation is always defined to be about the origin. After the rotation, you can move it back. – Matti P. Jun 28 '19 at 08:12
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With a two-dimensional,take (2, 1) as the center point and consider a transformation with a rotation angle of 45◦, then point (3, 3) tranformed into what point? – London Jack Jun 28 '19 at 08:18
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@London Jack, please refer to my answer. – PCeltide Jun 28 '19 at 08:19
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Where are you getting all of these exercises for which you keep asking for step-by-step instructions? They look like they’re all coming from an introductory linear algebra course. If so, then all of these step-by-step instructions that you want are in the course material. Similar problems have also been asked and answered many times over on this site. Have you tried searching for similar problem before posting your own? For instance, this exact problem is posed in this question from the handy list of related questions at right. – amd Jun 28 '19 at 17:45
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Moreover, this is fundamentally no different from this question that you asked only an hour before. – amd Jun 28 '19 at 17:51
1 Answers
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When $(2,1)$ is the center, $(3,3)$ will correspond to the point $(1,2)$.
Now, the tranformation rotates the bector by 45°.
Take the basis vectors $(1,0)$ and $(0,1)$.
$(1,0)$ will be transformed to $v = (-1/ \sqrt{2}, 1/ \sqrt{2})$ and $(0,1)$ will be transformed to $u= (1/ \sqrt{2}, 1/ \sqrt{2})$.
The transformation matrix will be $[v^T:u^T]= A$
So, the point $x = (1,2)$ will be transformed to,
$Ax= (1/ \sqrt{2}, 3/ \sqrt{2})$
PCeltide
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No you don't right? Because, the question has asked you where would (3,3) be transformed to. Once you shifted the origin, you consider different coordinates for the point, which itself is a transformation. So what you needed to do is compostion of transformation. You do not have to shift your origin back. – PCeltide Jun 28 '19 at 08:17
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@PratikApshinge So you're saying rotating $(3,3)$ about $(2,1)$ by 0 degrees would yield $(1,2)$? – Christoph Jun 28 '19 at 08:27
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@London Jack, if you had not wanted to solve with a matrix, please explain it in your question. Also, you should explain what you yourself have tried so we may know how to approach this question – PCeltide Jun 28 '19 at 08:30
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For $x=(1,2)$ you are applying a rotation about $(2,1)$ that takes $(3,3)=(2,1)+x$ to $(2,1)+Ax$. – Christoph Jun 28 '19 at 08:31
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I think, I explained myself wrong. Firstly, I haven't learned matrix,yet. I was asked to solve above question. However; I have learned rotation with just 90°,270°,180° degrees. Let me ,give example: – London Jack Jun 28 '19 at 10:24
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Ex: Find an image of the point (3, 2) that has undergone a clockwise rotation: a) about the origin at 90◦, b) about the origin at 180◦, and c) about the origin at 270◦. Write the notation to describe the rotation ..I can solve like this problem. I mean I don't know how to solve if degre other than 90°,270°,180°. – London Jack Jun 28 '19 at 10:25
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So, Question is again: With a two dimensional surface, if we take (2, 1) as the center point and consider a transformation with a rotation angle of 45◦, then point (3, 3) is transformed into which point? I am self-learner. I couldn't solve this problem. – London Jack Jun 28 '19 at 10:31