aortiH
2021-01-02
Answered

Find a transformation from u,v space to x,y,z space that takes the triangle $U=\left[\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\end{array}\right]$ to the triangle

$T=[(1,0),-2),(-1,2,0),(1,1,2)]$

You can still ask an expert for help

averes8

Answered 2021-01-03
Author has **92** answers

Step 1

The triangle$U=\left[\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\end{array}\right]$ and the triangle

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

Step 2

Consider$T(u,v)=({a}_{1}u+{b}_{1}v+{c}_{1},{a}_{2}u+{b}_{2}v+{c}_{2},{a}_{3}u+{b}_{3}v={c}_{3})$ .Obtain the transformation from u, v to x, y, z as

$T(0,0)=({a}_{1}\left(0\right)+{b}_{1}\left(0\right)+{c}_{1},{a}_{2}\left(0\right)+{b}_{2}\left(0\right)+{c}_{2},{a}_{3}\left(0\right)+{b}_{3}\left(0\right)+{c}_{3})$

$T(0,0)=({c}_{1},{c}_{2},{c}_{3})$

$T(0,0)=(1,0,-2)$

$\Rightarrow {c}_{1}=1,{c}_{2}=0,{c}_{3}=-2$

Step 3

Further evaluate as,

$T(1,0)=\left({a}_{1}\left(1\right)\right)+{b}_{1}\left(0\right)+{c}_{1},{a}_{2}\left(1\right)+{b}_{2}\left(0\right)+{c}_{2},{a}_{3}\left(1\right)+{b}_{2}\left(0\right)+{c}_{3})$

$T(0,0)=({a}_{1}+{c}_{1},{a}_{2}+{c}_{2},{a}_{3}+{c}_{3})$

$T(0,0)=(-1,2,0)$

$\Rightarrow {a}_{1}={c}_{1}=-1,{a}_{2}+{c}_{2}=2,{a}_{3}+{c}_{3}=0$

$\Rightarrow {a}_{1}=-2,{a}_{2}=2,{a}_{3}=2$

Step 4

Also evaluate,

$$###### Not exactly what you’re looking for?

The triangle

Step 2

Consider

Step 3

Further evaluate as,

Step 4

Also evaluate,

asked 2021-08-16

Discrete mathematics

If${x}_{1}=2,{x}_{n}=4{X}_{n-1}-4n\mathrm{\forall}n\ge 2$ .

Find the general term xn.

If

Find the general term xn.

asked 2021-08-11

Consider the following statement:

” A is a subset of B. Therefore A is a subset of P(B).”

This statement is incorrect as written. Assuming the first sentence is true, what is incorrect about the second sentence? State the second sentence correctly.

” A is a subset of B. Therefore A is a subset of P(B).”

This statement is incorrect as written. Assuming the first sentence is true, what is incorrect about the second sentence? State the second sentence correctly.

asked 2022-06-18

Consider the non-homogeneous linear recurrence relations ${a}_{n}=2{a}_{n-1}+{2}^{n}$ find all solutions.

I can show that ${a}_{n}^{(h)}$ characteristic equation $r-2=0\to {a}_{n}^{(h)}=\alpha {2}^{n}$.

But I'm stuck on ${a}_{n}^{(p)}$ characteristic equation $A{2}^{n}=2A{2}^{n-1}+{2}^{n}$

Simplifies to $-A=2$

$A=2A+2$

I can show that ${a}_{n}^{(h)}$ characteristic equation $r-2=0\to {a}_{n}^{(h)}=\alpha {2}^{n}$.

But I'm stuck on ${a}_{n}^{(p)}$ characteristic equation $A{2}^{n}=2A{2}^{n-1}+{2}^{n}$

Simplifies to $-A=2$

$A=2A+2$

asked 2021-08-21

Discrete math Question.

Suppose your friend makes the following English statement "If

asked 2021-08-15

asked 2022-06-25

Battleship placement proving that the number of battleships is divisible by 3

We have a grid of 6 columns and x number of rows. All battleships are three units long and can be placed like this: 1 or like this: 2

Where the entire grid is filled with ships with no square units left unfilled, I'm interested in showing that the number of ships positioned as seen in (2) must be divisible by 3; the number of ships placed upright is divisible by 3.

I noticed that the restriction on the number of columns to be 6 means that each column can only have up to two of the ships of type (1), I can show that the number of square units remaining is divisible by 3 but I know that this does not imply the actual number of ships is divisible by 3.

We have a grid of 6 columns and x number of rows. All battleships are three units long and can be placed like this: 1 or like this: 2

Where the entire grid is filled with ships with no square units left unfilled, I'm interested in showing that the number of ships positioned as seen in (2) must be divisible by 3; the number of ships placed upright is divisible by 3.

I noticed that the restriction on the number of columns to be 6 means that each column can only have up to two of the ships of type (1), I can show that the number of square units remaining is divisible by 3 but I know that this does not imply the actual number of ships is divisible by 3.

asked 2022-05-17

Existence of a path of length n/2 in every bipartite graph with $d(A,B)=1/2$

Claim: Let $G=A\cup B$ be a balanced bipartite graph with $e(A,B)\ge n/2$ then G has a path of length n/2.

I know about the erdos-gallai theorem that would net a path of length n/4. By noting that $d(G)=2E(G)/V(G)\ge {n}^{2}/4n$

I suspect that the condition of being bipartite forces the ecistence of a longer path, and I am yet unaware of such a result or a counterexample.

Part of my intuition is from the fact that considering a disjoint union of 2 copies of ${K}_{n/4-1,n/4}$ which are edge-maximal biparite graphs not containing such a path, we then have these two subgraphs and two yet to be connected vertices on A, also:

$2e({K}_{n/4-1,n/4})=\frac{{n}^{2}}{8}-\frac{n}{2}<{n}^{2}/8$

And adding any edge would form a path of the desired length. Any help on how to go about proving this, or a reference for such a resukt would be greatly appreciated.

Claim: Let $G=A\cup B$ be a balanced bipartite graph with $e(A,B)\ge n/2$ then G has a path of length n/2.

I know about the erdos-gallai theorem that would net a path of length n/4. By noting that $d(G)=2E(G)/V(G)\ge {n}^{2}/4n$

I suspect that the condition of being bipartite forces the ecistence of a longer path, and I am yet unaware of such a result or a counterexample.

Part of my intuition is from the fact that considering a disjoint union of 2 copies of ${K}_{n/4-1,n/4}$ which are edge-maximal biparite graphs not containing such a path, we then have these two subgraphs and two yet to be connected vertices on A, also:

$2e({K}_{n/4-1,n/4})=\frac{{n}^{2}}{8}-\frac{n}{2}<{n}^{2}/8$

And adding any edge would form a path of the desired length. Any help on how to go about proving this, or a reference for such a resukt would be greatly appreciated.