Chapter 3: Q17MP (page 186)

In Problems 14to 17, multiply matrices to find the resultant transformation. Caution: Be sure you are multiplying the matrices in the right order.
${\begin{matrix} x' = (2 x + y + 2 z) / 3 \\ y' = (x + 2 y - 2 z) / 3 \\ z' = (2 x - 2 y - z) / 3 \end{matrix} {\begin{matrix} x'' = (2 x' + y' + 2 z') / 3 \\ y'' = (- x' - 2 y' + 2 z') / 3 \\ z'' = (- 2 x' + 2 y' + z') / 3 \end{matrix}$

Short Answer

Expert verified

The resultant of the transformation for the multiplication of the given transformations is given below,

$x^{''} = y y^{''} = - x z^{''} = z$

Step by step solution

Given information

Given matrix in the question is,

\{\begin{matrix} x' = (2 x + y + 2 z) / 3 \\ y' = (x + 2 y - 2 z) / 3 \\ z' = (2 x - 2 y - z) / 3 \end{matrix} \{\begin{matrix} x'' = (2 x' + y' + 2 z') / 3 \\ y'' = (- x' - 2 y' + 2 z') / 3 \\ z'' = (- 2 x' + 2 y' + z') / 3 \end{matrix}

Transformation matrix.

A transformation matrix is a matrix that, through the process of matrix multiplication, turns one vector into another vector.

Find the resultant transformation for the multiplication of the matrices

Express the transformation in the matrix for the first transformation:

$[\begin{matrix} x' \\ y' \\ z' \end{matrix}] [\begin{matrix} \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \\ \frac{1}{\sqrt{2}} & 0 & - \frac{1}{\sqrt{2}} \\ - \frac{1}{2} & \frac{1}{\sqrt{2}} & - \frac{1}{2} \end{matrix}]$

$[\begin{matrix} x'' \\ y'' \\ z'' \end{matrix}] = [\begin{matrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ - \frac{1}{2} & - \frac{1}{\sqrt{2}} & \frac{1}{2} \\ \frac{1}{2} & - \frac{1}{\sqrt{2}} & - \frac{1}{2} \end{matrix}] [\begin{matrix} x' \\ y' \\ z' \end{matrix}] [\begin{matrix} x'' \\ y'' \\ z'' \end{matrix}] = [\begin{matrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ - \frac{1}{2} & - \frac{1}{\sqrt{2}} & \frac{1}{2} \\ \frac{1}{2} & - \frac{1}{\sqrt{2}} & - \frac{1}{2} \end{matrix}] [\begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{2} \\ \frac{1}{\sqrt{2}} & 0 & - \frac{1}{\sqrt{2}} \\ - \frac{1}{2} & \frac{1}{\sqrt{2}} & - \frac{1}{2} \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] [\begin{matrix} x'' \\ y'' \\ z'' \end{matrix}] = [\begin{matrix} \frac{1}{2 \sqrt{2}} + 0 - \frac{1}{2 \sqrt{2}} & \frac{1}{2} + 0 + \frac{1}{2} & \frac{1}{2 \sqrt{2}} + 0 - \frac{1}{2 \sqrt{2}} \\ - \frac{1}{4} - \frac{1}{2} - \frac{1}{4} & - \frac{1}{2 \sqrt{2}} + 0 + \frac{1}{2 \sqrt{2}} & - \frac{1}{4} + \frac{1}{2} - \frac{1}{4} \\ \frac{1}{4} - \frac{1}{2} + \frac{1}{4} & \frac{1}{2 \sqrt{2}} + 0 - \frac{1}{2 \sqrt{2}} & \frac{1}{4} + \frac{1}{2} + \frac{1}{4} \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] [\begin{matrix} x'' \\ y'' \\ z'' \end{matrix}] = [\begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}]$

Therefore, the resultant of the transformation for the multiplication of the given transformations of the equation 1 and 2 is evaluated as:

$x^{''} = y y^{''} = - x z^{''} = z$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Given information

Transformation matrix.

Find the resultant transformation for the multiplication of the matrices

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Circular Motion and Gravitation

Electric Charge Field and Potential

Further Mechanics and Thermal Physics

Famous Physicists

Wave Optics

Study anywhere. Anytime. Across all devices.

In Problems 14to 17, multiply matrices to find the resultant transformation. Caution: Be sure you are multiplying the matrices in the right order.{x'=2x+y+2z/3y'=x+2y-2z/3z'=2x-2y-z/3{x''=2x'+y'+2z'/3y''=-x'-2y'+2z'/3z''=-2x'+2y'+z'/3

Short Answer

Step by step solution

Given information

Transformation matrix.

Find the resultant transformation for the multiplication of the matrices

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Circular Motion and Gravitation

Electric Charge Field and Potential

Further Mechanics and Thermal Physics

Famous Physicists

Wave Optics

Study anywhere. Anytime. Across all devices.