Which expression consists of *exactly* two terms when simplified?

(7 + 3)*x*

(7 + 3), x

11(*x* + 7)

11(, x, + 7)

12*x* + 5(*y* + 8)

12, x, + 5(, y, + 8)

(3*x* + 8*y* ) + 4*z*

Endiyah Neal
2022-08-12

Which expression consists of *exactly* two terms when simplified?

(7 + 3)*x*

(7 + 3), x

11(*x* + 7)

11(, x, + 7)

12*x* + 5(*y* + 8)

12, x, + 5(, y, + 8)

(3*x* + 8*y* ) + 4*z*

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asked 2021-02-25

Determine if $\{\left[\begin{array}{c}2\\ -4\\ 1\end{array}\right],\left[\begin{array}{c}-3\\ 5\\ -1\end{array}\right]\}$ is a basis for

$Col\left[\begin{array}{c}1\\ -3\\ 1\end{array}\right],\left[\begin{array}{c}-3\\ 7\\ -2\end{array}\right]$

asked 2022-09-05

Given ax+by+c=0, what is the set of all operations on this equation that do not alter the plotted line?

Operations such as $f(x)+a-a$, are obvious candidates for such a set. However, e.g., for the line y=−x, it seems to me to be non-trivial that ${x}^{3}+{y}^{3}=0$ will plot the same line but ${x}^{2}+{y}^{2}=0$ won't. Translation between coordinate systems also seems to be a non-trivial example. Is there any way to designate such a set? (Could this be generalized to other types of curves?)

The following are some more thoughts on the question:

It would be interesting in order to find alternate equation forms that might make more clear certain properties of a curve. For instance, $\frac{x}{a}+\frac{x}{b}=1$ makes immediately obvious the abscissa and ordinate at origin. But we know that under some types of algebra, $ax+by+c=0$ might fail to be represented $\frac{x}{a}+\frac{x}{b}=1$. So we're lead to think that these two equations plot a line by virtue of legitimate operations between them.

The equation of a plane also seems to be nicely related to the general form of a line, if ${r}_{0}=({x}_{0},{y}_{0})$ and r=(x,y) are two vectors pointing to the plane and the normal is $n=({n}_{x},{n}_{y})$. If $\circ $ between vectors is the dot product, $(x-{x}_{0},y-{y}_{0})\circ n=(x-{x}_{0})\ast {n}_{x}+(y-{y}_{0})\ast {n}_{y}={n}_{x}\ast x+{n}_{y}\ast y-({x}_{0}{n}_{x}+{y}_{0}{n}_{y})=a\ast x+b\ast y+c=0$

The idea is to be able to see how the form of an equation can be altered, not the content of the variables. It seems odd to me that very complicated equations could have the same plotted curve as simple forms, but that this property wouldn't appear by virtue of the equation themselves, or the set of valid operations on this equation. This might seem weird, but say it is never immediately obvious that ax+by+c=0 plots a line, or ${x}^{2}+{y}^{2}={r}^{2}$ plots a circle, unless we actually do the plotting, and ax+by+c=0 seems way less fundamental than y=mx+b.

Note that in the case of a circle, we have the pythagorean theorem that seems to be its clearest representation with the methods of analytic geometry, and the moment an equation can be said to share some sort of operation set with the pythagorean theorem, we know we're speaking of a circle. It seems that if we could somehow draw the operation set of a circle, we would get something like the pythagorean theorem, and that this operation set gets somehow deformed in order to give a representation onto the cartesian plane. For a translated circle with center (h,k), ${x}^{2}-2xh+{h}^{2}+{y}^{2}-2yk+{k}^{2}={r}^{2}$ means absolutely nothing to us, but the form $(x-h{)}^{2}+(y-k{)}^{2}={r}^{2}$ is clear as day.

Operations such as $f(x)+a-a$, are obvious candidates for such a set. However, e.g., for the line y=−x, it seems to me to be non-trivial that ${x}^{3}+{y}^{3}=0$ will plot the same line but ${x}^{2}+{y}^{2}=0$ won't. Translation between coordinate systems also seems to be a non-trivial example. Is there any way to designate such a set? (Could this be generalized to other types of curves?)

The following are some more thoughts on the question:

It would be interesting in order to find alternate equation forms that might make more clear certain properties of a curve. For instance, $\frac{x}{a}+\frac{x}{b}=1$ makes immediately obvious the abscissa and ordinate at origin. But we know that under some types of algebra, $ax+by+c=0$ might fail to be represented $\frac{x}{a}+\frac{x}{b}=1$. So we're lead to think that these two equations plot a line by virtue of legitimate operations between them.

The equation of a plane also seems to be nicely related to the general form of a line, if ${r}_{0}=({x}_{0},{y}_{0})$ and r=(x,y) are two vectors pointing to the plane and the normal is $n=({n}_{x},{n}_{y})$. If $\circ $ between vectors is the dot product, $(x-{x}_{0},y-{y}_{0})\circ n=(x-{x}_{0})\ast {n}_{x}+(y-{y}_{0})\ast {n}_{y}={n}_{x}\ast x+{n}_{y}\ast y-({x}_{0}{n}_{x}+{y}_{0}{n}_{y})=a\ast x+b\ast y+c=0$

The idea is to be able to see how the form of an equation can be altered, not the content of the variables. It seems odd to me that very complicated equations could have the same plotted curve as simple forms, but that this property wouldn't appear by virtue of the equation themselves, or the set of valid operations on this equation. This might seem weird, but say it is never immediately obvious that ax+by+c=0 plots a line, or ${x}^{2}+{y}^{2}={r}^{2}$ plots a circle, unless we actually do the plotting, and ax+by+c=0 seems way less fundamental than y=mx+b.

Note that in the case of a circle, we have the pythagorean theorem that seems to be its clearest representation with the methods of analytic geometry, and the moment an equation can be said to share some sort of operation set with the pythagorean theorem, we know we're speaking of a circle. It seems that if we could somehow draw the operation set of a circle, we would get something like the pythagorean theorem, and that this operation set gets somehow deformed in order to give a representation onto the cartesian plane. For a translated circle with center (h,k), ${x}^{2}-2xh+{h}^{2}+{y}^{2}-2yk+{k}^{2}={r}^{2}$ means absolutely nothing to us, but the form $(x-h{)}^{2}+(y-k{)}^{2}={r}^{2}$ is clear as day.

asked 2022-09-04

Surprisingly elementary and direct proofs

What are some examples of theorems, whose first proof was quite hard and sophisticated, perhaps using some other deep theorems of some theory, before years later surprisingly a quite elementary, direct, perhaps even short proof has been found?

A related question is MO/24913, which deals with hard theorems whose proofs were simplified by the development of more sophisticated theories. But I would like to see examples where this wasn't necessary, but rather the theory turned out to be superfluous as for the proof of the theorem. I expect that this didn't happen so often. [Ok after reading all the answers, it obviously happened all the time!]

What are some examples of theorems, whose first proof was quite hard and sophisticated, perhaps using some other deep theorems of some theory, before years later surprisingly a quite elementary, direct, perhaps even short proof has been found?

A related question is MO/24913, which deals with hard theorems whose proofs were simplified by the development of more sophisticated theories. But I would like to see examples where this wasn't necessary, but rather the theory turned out to be superfluous as for the proof of the theorem. I expect that this didn't happen so often. [Ok after reading all the answers, it obviously happened all the time!]

asked 2022-09-04

Deriving the distance of closest approach between ellipsoid and line (prev. "equation of a 3-dimensional line in spherical coordinates")

Currently trying to solve a problem of calculating the smallest distance between a given ellipsoid centered on the coordinate system starting point and a given line (located... somewhere).

After chasing a few promising but non-functional methods I have settled on trying to use spherical coordintates - to determine the formula for said distance by determining the formula of the distance of the ellipsoid from the center and doing the same for the line, consequently subtracting the two, and using gradient descent on the resulting function to approach a minimum (hopefully the global one).

However, while that makes the ellipsoid calculations easy, I have found no consise way of determining a line in spherical coordinates in equation form. I have found the old questions on similar topics proposing use of Euler angles and the like, but that does not seem to be the solution (possibly because I haven't managed to appreciate it). So, asking here - is there any way to derive an equation for a line in 3-dimensional space in spherical coordinates?

Alternate methods for the task at hand are appreciated too - for the record, my previous lead was using a cyllindrical coordinate system with the line as its X axis, but the resulting formula for the ellipsoid turned out to be bogus.

Edit: may have figured out a solution for the bigger problem that does not rely on the spherical equation - see my own answer to the question. Title changed accordingly.

Edit 2: Scratch that. Fell into the same trap again; that solution is not going to work.

Currently trying to solve a problem of calculating the smallest distance between a given ellipsoid centered on the coordinate system starting point and a given line (located... somewhere).

After chasing a few promising but non-functional methods I have settled on trying to use spherical coordintates - to determine the formula for said distance by determining the formula of the distance of the ellipsoid from the center and doing the same for the line, consequently subtracting the two, and using gradient descent on the resulting function to approach a minimum (hopefully the global one).

However, while that makes the ellipsoid calculations easy, I have found no consise way of determining a line in spherical coordinates in equation form. I have found the old questions on similar topics proposing use of Euler angles and the like, but that does not seem to be the solution (possibly because I haven't managed to appreciate it). So, asking here - is there any way to derive an equation for a line in 3-dimensional space in spherical coordinates?

Alternate methods for the task at hand are appreciated too - for the record, my previous lead was using a cyllindrical coordinate system with the line as its X axis, but the resulting formula for the ellipsoid turned out to be bogus.

Edit: may have figured out a solution for the bigger problem that does not rely on the spherical equation - see my own answer to the question. Title changed accordingly.

Edit 2: Scratch that. Fell into the same trap again; that solution is not going to work.

asked 2022-07-02

Find the equation of the line that satisfies the given conditions:

Goes through $(-1,4)$, slope $-9$

Goes through $(-1,4)$, slope $-9$

asked 2021-01-02

To Sketch: The points on three - axis Cartesian coordinate system.

Point A:$\left[\begin{array}{ccc}-6& \text{}2& \text{}1\end{array}\right]$

Point B:$\left[\begin{array}{ccc}-6& \text{}2& \text{}1\end{array}\right]$

Point C:$\left[\begin{array}{ccc}5& \text{}-3& \text{}-6\end{array}\right]$

Point A:

Point B:

Point C:

asked 2021-01-23

To find: the distance between the lakes to the nearest mile.

Given:

(26,15) and (9,20)

Given:

(26,15) and (9,20)