Derive the matrix that represents a pure rotation about the (i) y-axis , (ii) z-axis and (iii) x-axis of the reference frame with the help of diagram.

glasskerfu
2021-02-20
Answered

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Alannej

Answered 2021-02-21
Author has **104** answers

Step 1:- Introduction:-

A basic rotation also called elemental rotation is a rotation about one of the axes of a co-ordinate system.

Step 2:- Calculation:-

The Following three basic rotation matrices rotate vectors by an angle$\theta $ about x, y, z axes , in three dimensions which codifies their alternating signs.

${R}_{x}(\theta )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\theta & -\mathrm{sin}\theta \\ 0& \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$

${R}_{y}(\theta )=\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& \mathrm{sin}\theta \\ 0& 1& 0\\ -\mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]$

${R}_{z}(\theta )=\left[\begin{array}{ccc}\mathrm{cos}\theta & -\mathrm{sin}\theta & 0\\ \mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right]$

A basic rotation also called elemental rotation is a rotation about one of the axes of a co-ordinate system.

Step 2:- Calculation:-

The Following three basic rotation matrices rotate vectors by an angle

Jeffrey Jordon

Answered 2022-01-29
Author has **2313** answers

Answer is given below (on video)

asked 2021-02-08

Let B be a 4x4 matrix to which we apply the following operations:

1. double column 1,

2. halve row 3,

3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

1. double column 1,

2. halve row 3,

3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

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$\text{Basis}=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

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Degrees of Sums and Products of Polynomials

Make up several pairs of polynomials, then calculate the sum and product of each pair. On the basis of your experiments and observations, answer the following questions.

a) How is the degree of the product related to the degrees of the original polynomials?

b) How is the degree of the sum related to the degrees of the original polynomials?

c) Test your conclusions by finding the sum and product of the following polynomials:

$2{x}^{3}+x-3$ and $-2{x}^{3}-x+7$

Make up several pairs of polynomials, then calculate the sum and product of each pair. On the basis of your experiments and observations, answer the following questions.

a) How is the degree of the product related to the degrees of the original polynomials?

b) How is the degree of the sum related to the degrees of the original polynomials?

c) Test your conclusions by finding the sum and product of the following polynomials:

asked 2021-12-30

Any trick for evaluating $(\frac{\sqrt{3}}{2}\mathrm{cos}\theta +\frac{i}{2}\mathrm{sin}\left(\theta \right))}^{7$ ?

Expressions of the form$(a\mathrm{cos}\left(\theta \right)+bi\mathrm{sin}\left(\theta \right))}^{n$ come up from time to time in applications of complex analysis, but to my knowledge the De Moivre's formula can only be applied with a=b. Is there some trick to deal with the case of $a\ne b$ , for example when the expression is $(\frac{\sqrt{3}}{2}\mathrm{cos}\theta +\frac{i}{2}\mathrm{sin}\theta )}^{7$ ?

Expressions of the form