What Are The Coordinates Of \[$ V \$\] After The Transformation \[$\left(T_{\langle 3,-2 \rangle} \cdot D_5\right)(\triangle TUV)\$\] For \[$ T(-1,-1), U(-1,2), V(2,1) \$\]? Coordinates Of \[$ V \$\] = \[$\square\$\]

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Understanding the Transformation

The given transformation involves two main components: a translation {T_{\langle 3,-2 \rangle}$] and a dilation [D_5\$}. To find the coordinates of point {V$] after the transformation, we need to understand how each component affects the point.

Translation

A translation is a transformation that moves a point or a figure from one location to another without changing its size or shape. In this case, the translation [T⟨3,−2⟩$]movesthepoint\[T_{\langle 3,-2 \rangle}\$] moves the point \[V$] by [3$]unitstotherightand\[3\$] units to the right and \[-2$] units down.

Applying the Translation

To apply the translation, we add the translation vector [⟨3,−2⟩$]tothecoordinatesofpoint\[\langle 3,-2 \rangle\$] to the coordinates of point \[V$]. The original coordinates of point [V$]are\[V\$] are \[\left(2,1\right)$]. Adding the translation vector, we get:

[$\left(2+3,1-2\right) = \left(5,-1\right)$]

So, after the translation, the coordinates of point [V$]are\[V\$] are \[\left(5,-1\right)$].

Dilation

A dilation is a transformation that changes the size of a figure. In this case, the dilation [$D_5$] scales the figure by a factor of [5$].Thismeansthatthecoordinatesofpoint\[5\$]. This means that the coordinates of point \[V$] will be multiplied by [$5$].

Applying the Dilation

To apply the dilation, we multiply the coordinates of point [$V$] by the scale factor [5$].Thecoordinatesofpoint\[5\$]. The coordinates of point \[V$] after the translation are [$\left(5,-1\right)$]. Multiplying by the scale factor, we get:

[$\left(5 \times 5, -1 \times 5\right) = \left(25,-5\right)$]

So, after the dilation, the coordinates of point [V$]are\[V\$] are \[\left(25,-5\right)$].

Conclusion

In conclusion, the coordinates of point [V$]afterthetransformation\[V\$] after the transformation \[\left(T_{\langle 3,-2 \rangle} \cdot D_5\right)(\triangle TUV)$] for [T(-1,-1), U(-1,2), V(2,1) \$} are {\left(25,-5\right)$].

Step-by-Step Solution

Here's a step-by-step solution to the problem:

  1. Apply the translation [T⟨3,−2⟩$]tothecoordinatesofpoint\[T_{\langle 3,-2 \rangle}\$] to the coordinates of point \[V$]. Add the translation vector [⟨3,−2⟩$]tothecoordinatesofpoint\[\langle 3,-2 \rangle\$] to the coordinates of point \[V$].
  2. Apply the dilation [D5$]tothecoordinatesofpoint\[D_5\$] to the coordinates of point \[V$]. Multiply the coordinates of point [$V$] by the scale factor [$5$].
  3. The final coordinates of point [$V$] are the result of applying both transformations.

Example

Let's consider an example to illustrate the transformation. Suppose we have a triangle [△TUV$]withvertices\[\triangle TUV\$] with vertices \[T(-1,-1), U(-1,2), V(2,1)$]. We want to apply the transformation [$\left(T_{\langle 3,-2 \rangle} \cdot D_5\right)(\triangle TUV)$] to the triangle.

Applying the Transformation

To apply the transformation, we first apply the translation [T⟨3,−2⟩$]totheverticesofthetriangle.Then,weapplythedilation\[T_{\langle 3,-2 \rangle}\$] to the vertices of the triangle. Then, we apply the dilation \[D_5$] to the translated vertices.

Translated Vertices

After applying the translation, the vertices of the triangle become:

[$T'(-1+3,-1-2) = T'(2,-3)$]

[$U'(-1+3,2-2) = U'(2,0)$]

[$V'(-1+3,1-2) = V'(2,-1)$]

Dilated Vertices

After applying the dilation, the translated vertices become:

[$T''(2 \times 5,-3 \times 5) = T''(10,-15)$]

[$U''(2 \times 5,0 \times 5) = U''(10,0)$]

[$V''(2 \times 5,-1 \times 5) = V''(10,-5)$]

Final Coordinates

The final coordinates of the vertices of the triangle are:

[$T''(10,-15)$]

[$U''(10,0)$]

[$V''(10,-5)$]

Conclusion

In conclusion, the coordinates of point [V$]afterthetransformation\[V\$] after the transformation \[\left(T_{\langle 3,-2 \rangle} \cdot D_5\right)(\triangle TUV)$] for [T(-1,-1), U(-1,2), V(2,1) \$} are {\left(25,-5\right)$].

Frequently Asked Questions

Q: What is the translation vector?

A: The translation vector is [$\langle 3,-2 \rangle$].

Q: What is the scale factor for the dilation?

A: The scale factor for the dilation is [$5$].

Q: How do I apply the translation to the coordinates of point [$V$]?

A: To apply the translation, add the translation vector [⟨3,−2⟩$]tothecoordinatesofpoint\[\langle 3,-2 \rangle\$] to the coordinates of point \[V$].

Q: How do I apply the dilation to the coordinates of point [$V$]?

A: To apply the dilation, multiply the coordinates of point [$V$] by the scale factor [$5$].

Q: What are the final coordinates of point [$V$] after the transformation?

A: The final coordinates of point [V$]afterthetransformationare\[V\$] after the transformation are \[\left(25,-5\right)$].

Q: What is the purpose of the translation [$T_{\langle 3,-2 \rangle}$] in the transformation?

A: The purpose of the translation [T⟨3,−2⟩$]istomovethepoint\[T_{\langle 3,-2 \rangle}\$] is to move the point \[V$] by [3$]unitstotherightand\[3\$] units to the right and \[-2$] units down.

Q: How does the dilation [D5$]affectthecoordinatesofpoint\[D_5\$] affect the coordinates of point \[V$]?

A: The dilation [D5$]scalesthecoordinatesofpoint\[D_5\$] scales the coordinates of point \[V$] by a factor of [5$].Thismeansthatthex−coordinateandy−coordinateofpoint\[5\$]. This means that the x-coordinate and y-coordinate of point \[V$] are multiplied by [$5$].

Q: What are the final coordinates of point [$V$] after the transformation?

A: The final coordinates of point [V$]afterthetransformationare\[V\$] after the transformation are \[\left(25,-5\right)$].

Q: How do I apply the translation to the coordinates of point [$V$]?

A: To apply the translation, add the translation vector [⟨3,−2⟩$]tothecoordinatesofpoint\[\langle 3,-2 \rangle\$] to the coordinates of point \[V$].

Q: How do I apply the dilation to the coordinates of point [$V$]?

A: To apply the dilation, multiply the coordinates of point [$V$] by the scale factor [$5$].

Q: What is the effect of the translation and dilation on the shape of the triangle [$\triangle TUV$]?

A: The translation and dilation do not change the shape of the triangle [$\triangle TUV$]. However, they do change the size and position of the triangle.

Q: Can I apply the translation and dilation in a different order?

A: Yes, you can apply the translation and dilation in a different order. However, the final result will be the same.

Q: How do I determine the coordinates of point [$V$] after the transformation?

A: To determine the coordinates of point [V$]afterthetransformation,youneedtoapplythetranslationanddilationtotheoriginalcoordinatesofpoint\[V\$] after the transformation, you need to apply the translation and dilation to the original coordinates of point \[V$].

Q: What are the key steps in applying the transformation?

A: The key steps in applying the transformation are:

  1. Apply the translation to the coordinates of point [$V$].
  2. Apply the dilation to the translated coordinates of point [$V$].

Q: Can I use the transformation to solve real-world problems?

A: Yes, you can use the transformation to solve real-world problems. For example, you can use the transformation to model the movement of objects in a coordinate plane.

Q: How do I visualize the transformation?

A: You can visualize the transformation by drawing a diagram of the triangle [$\triangle TUV$] and applying the translation and dilation to the diagram.

Q: What are the limitations of the transformation?

A: The limitations of the transformation are:

  1. The transformation only works for points in a coordinate plane.
  2. The transformation does not change the shape of the triangle [$\triangle TUV$].

Q: Can I apply the transformation to other shapes?

A: Yes, you can apply the transformation to other shapes. However, the transformation may not work for all shapes.

Q: How do I determine the scale factor for the dilation?

A: To determine the scale factor for the dilation, you need to know the ratio of the new size to the original size of the shape.

Q: What are the key concepts in the transformation?

A: The key concepts in the transformation are:

  1. Translation
  2. Dilation
  3. Coordinate plane

Q: Can I use the transformation to solve problems in other fields?

A: Yes, you can use the transformation to solve problems in other fields. For example, you can use the transformation to model the movement of objects in physics or engineering.

Q: How do I apply the transformation to a 3D object?

A: To apply the transformation to a 3D object, you need to apply the translation and dilation to each point of the object in 3D space.

Q: What are the challenges in applying the transformation?

A: The challenges in applying the transformation are:

  1. Understanding the translation and dilation
  2. Applying the transformation to complex shapes
  3. Visualizing the transformation in 3D space

Q: Can I use the transformation to solve problems in computer graphics?

A: Yes, you can use the transformation to solve problems in computer graphics. For example, you can use the transformation to model the movement of objects in a video game.

Q: How do I determine the coordinates of point [$V$] after the transformation in 3D space?

A: To determine the coordinates of point [V$]afterthetransformationin3Dspace,youneedtoapplythetranslationanddilationtotheoriginalcoordinatesofpoint\[V\$] after the transformation in 3D space, you need to apply the translation and dilation to the original coordinates of point \[V$] in 3D space.

Q: What are the key applications of the transformation?

A: The key applications of the transformation are:

  1. Computer graphics
  2. Physics
  3. Engineering

Q: Can I use the transformation to solve problems in other fields?

A: Yes, you can use the transformation to solve problems in other fields. For example, you can use the transformation to model the movement of objects in economics or finance.

Q: How do I apply the transformation to a real-world problem?

A: To apply the transformation to a real-world problem, you need to:

  1. Understand the problem
  2. Identify the key concepts in the problem
  3. Apply the transformation to the problem

Q: What are the limitations of the transformation in real-world applications?

A: The limitations of the transformation in real-world applications are:

  1. The transformation only works for points in a coordinate plane.
  2. The transformation does not change the shape of the object.

Q: Can I use the transformation to solve problems in other areas of mathematics?

A: Yes, you can use the transformation to solve problems in other areas of mathematics. For example, you can use the transformation to model the movement of objects in calculus or differential equations.

Q: How do I determine the coordinates of point [$V$] after the transformation in other areas of mathematics?

A: To determine the coordinates of point [V$]afterthetransformationinotherareasofmathematics,youneedtoapplythetranslationanddilationtotheoriginalcoordinatesofpoint\[V\$] after the transformation in other areas of mathematics, you need to apply the translation and dilation to the original coordinates of point \[V$] in the specific area of mathematics.

Q: What are the key concepts in the transformation in other areas of mathematics?

A: The key concepts in the transformation in other areas of mathematics are:

  1. Translation
  2. Dilation
  3. Coordinate plane

Q: Can I use the transformation to solve problems in other areas of science?

A: Yes, you can use the transformation to solve problems in other areas of science. For example, you can use the transformation to model the movement of objects in physics or engineering.

Q: How do I apply the transformation to a real-world problem in other areas of science?

A: To apply the transformation to a real-world problem in other areas of science, you need to:

  1. Understand the problem
  2. Identify the key concepts in the problem
  3. Apply the transformation to the problem

Q: What are the limitations of the transformation in other areas of science?

A: The limitations of the transformation in other areas of science are:

  1. The transformation only works for points in a coordinate plane.
  2. The transformation does not change the shape of the object.

Q: Can I use the transformation to solve problems in other areas of technology?

A: Yes, you can use the transformation to solve problems in other areas of technology. For example, you can use the transformation to model the movement of objects in computer graphics or video games.

Q: How do I apply the transformation to a real-world problem in other areas of technology?

A: To apply the transformation to a real-world problem in other areas of technology, you need to:

  1. Understand the problem
  2. Identify the key concepts in the problem
  3. Apply the transformation to the problem

Q: What are the limitations of the transformation in other areas of technology?

A: The limitations of the transformation in other areas of technology are:

  1. The transformation only works for points in a coordinate plane.
  2. The transformation does not change the shape of the object.