Khaleesi Herbert

2021-02-19

(a) The new coordinates geometrically if X represents the point $(0,\text{}\sqrt{2})$ m and this

point is rotated about the origin${45}^{\circ}$ clockwise and then translated 2 units to the right and 3 units upward.

(b) The value of$Y=ABX,$ and explain the result.

(c) If ABX equal to BAX. Interpret the resul.

(d) A matrix that translate Y back to X

point is rotated about the origin

(b) The value of

(c) If ABX equal to BAX. Interpret the resul.

(d) A matrix that translate Y back to X

d2saint0

Skilled2021-02-20Added 89 answers

Given:

The matrices

$A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 0& 0& 1\end{array}\right],\text{}B=\left[\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right]$

Concept used:

When a point is rotated an angle theta clockwise about the origin, the transforming matrix is

$\left[\begin{array}{ccc}\mathrm{cos}\theta & \mathrm{sin}\theta & 0\\ -\mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right]$

To translate a point (x, y) horizontallyh units and vertically k units, we use the transformation matrix

$\left[\begin{array}{ccc}1& 0& h\\ 0& 1& k\\ 0& 0& 1\end{array}\right]$

Calculation:

(a) Let the point$(0,\text{}\sqrt{2})$ be represented by column matrix

$X=\left[\begin{array}{c}1\\ \sqrt{2}\\ 1\end{array}\right]$

When it is rotated${45}^{\circ}$ about the origin, then we have

$\left[\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}0\\ \sqrt{2}\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

So after rotating${45}^{\circ}$ about the origin, the new coordinates are (1, 1).

Now we need to translate this point 2 units to the right and 3 units upward, so using above concept we have

$\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}3\\ 4\\ 1\end{array}\right]$

Hence after rotating${45}^{\circ}$ about origin and then translating 2 units to the right and 3 units upward, we reach to the point (3, 4).

(b) Now to compure$Y=ABX,$ we have

$Y=ABX=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ 1& 0& 1\end{array}\right]\left[\begin{array}{c}0\\ \sqrt{2}\\ 1\end{array}\right]$

$=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

$=\left[\begin{array}{c}3\\ 4\\ 1\end{array}\right]$

It is same as the above transformations.

Hence the first matrix A represents the translation of 2 units the right and 3 units upward.

And the matrix B represent the rotation of${45}^{\circ}$ clockwise about the origin of the point X.

(c) Now to check if ABX equal to BAX, we compute BAX first. So we have

$BAX=\left[\begin{array}{ccc}\frac{1}{\sqrt{1}}& \frac{1}{\sqrt{2}}& 0\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}2\\ 3\text{}+\text{}\sqrt{2}\\ 1\end{array}\right]\left[\begin{array}{c}\frac{5\text{}+\text{}\sqrt{2}}{\sqrt{2}}\\ \frac{1\text{}+\text{}\sqrt{2}}{\sqrt{2}}\\ 1\end{array}\right]$

We can see that ABX is not equal to BAX. It implies that if we translate the point X, 2 units to the right and 3 units upward first and then rotate about the origin${45}^{\circ}$ clockwise we will not reach to the same point as we get by first rotating about the origin and then translating.

(d) To find the matrix that translate Y back to X, we use the concept

$Y=ABX$

The matrices

Concept used:

When a point is rotated an angle theta clockwise about the origin, the transforming matrix is

To translate a point (x, y) horizontallyh units and vertically k units, we use the transformation matrix

Calculation:

(a) Let the point

When it is rotated

So after rotating

Now we need to translate this point 2 units to the right and 3 units upward, so using above concept we have

Hence after rotating

(b) Now to compure

It is same as the above transformations.

Hence the first matrix A represents the translation of 2 units the right and 3 units upward.

And the matrix B represent the rotation of

(c) Now to check if ABX equal to BAX, we compute BAX first. So we have

We can see that ABX is not equal to BAX. It implies that if we translate the point X, 2 units to the right and 3 units upward first and then rotate about the origin

(d) To find the matrix that translate Y back to X, we use the concept

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1)?

How to expand and simplify $2(3x+4)-3(4x-5)$?

Find an equation equivalent to ${x}^{2}-{y}^{2}=4$ in polar coordinates.

How to graph $r=5\mathrm{sin}\theta$?

How to find the length of a curve in calculus?

When two straight lines are parallel their slopes are equal.

A)True;

B)FalseIntegration of 1/sinx-sin2x dx

Converting percentage into a decimal. $8.5\%$

Arrange the following in the correct order of increasing density.

Air

Oil

Water

BrickWhat is the exact length of the spiraling polar curve $r=5{e}^{2\theta}$ from 0 to $2\pi$?

What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?

What is the slope of the tangent line of $r=-2\mathrm{sin}\left(3\theta \right)-12\mathrm{cos}\left(\frac{\theta}{2}\right)$ at $\theta =\frac{-\pi}{3}$?

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?

Use the summation formulas to rewrite the expression $\Sigma \frac{2i+1}{{n}^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000.

How to calculate the right hand and left hand riemann sum using 4 sub intervals of f(x)= 3x on the interval [1,5]?