I have something starting at (50, 10) it then rotates counter clockwise by 30 degrees, around the point at (50, 0), essentially mapping out an arc of a circle. How do I find the point it now lies on?

Yesenia Sherman
2022-06-29
Answered

I have something starting at (50, 10) it then rotates counter clockwise by 30 degrees, around the point at (50, 0), essentially mapping out an arc of a circle. How do I find the point it now lies on?

You can still ask an expert for help

Colin Moran

Answered 2022-06-30
Author has **21** answers

You apply a rotation matrix to the vector from the center of rotation to the point that is rotating. Here the vector is $(50-50,10-0)=(0,10)$. Then you multiply that by $\left[\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$, getting

$\left[\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 10\end{array}\right]=\left[\begin{array}{c}10\mathrm{sin}{30}^{\circ}\\ 10\mathrm{cos}{30}^{\circ}\end{array}\right]=\left[\begin{array}{c}5\\ 5\sqrt{3}\end{array}\right]$

and add that to the center, getting $(50+5,0+5\sqrt{3})=(55,5\sqrt{3})$

$\left[\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 10\end{array}\right]=\left[\begin{array}{c}10\mathrm{sin}{30}^{\circ}\\ 10\mathrm{cos}{30}^{\circ}\end{array}\right]=\left[\begin{array}{c}5\\ 5\sqrt{3}\end{array}\right]$

and add that to the center, getting $(50+5,0+5\sqrt{3})=(55,5\sqrt{3})$

asked 2022-05-07

I am attempting to determine the locus of z for $\mathrm{arg}({z}^{2}+1)$. I do know that since ${z}^{2}+1=(z+i)(z-i)$, I could write $\mathrm{arg}({z}^{2}+1)=\mathrm{arg}(z+i)+\mathrm{arg}(z-i)$ but I don’t really know how to figure out the locus of the points from here. How should I approach this? Is there a relationship with the difference in two arguments (arc of a circle)? Thank you!

asked 2022-07-17

Let segment gh be a chord of a circle $\omega $ which is not a diameter, and let n be a fixed point on gh. For which point b on arc gh is the length n minimized?

asked 2022-06-21

My teacher said that the central angles of a circle are equal to the measure of the arc, but I don't understand on how this could possibly work.

Can someone please explain how this is possible?

Can someone please explain how this is possible?

asked 2022-07-21

Find a harmonic function on $|z|<1$ with value 1 on an arc of $|z|=1$ and zero on the rest.

Of course, if allow the arc with value 1 to be the whole $|z|=1$ circle it would be trivial.

Without that, how to construct such a function, I'm a bit lost. I suppose if I can find a holomorphic function $f(z)$ whose real part full fill that it shall work, but playing around with the normal functions such as ${e}^{x}$, $\mathrm{sin}x$ etc, I can't find one with such property.

Of course, if allow the arc with value 1 to be the whole $|z|=1$ circle it would be trivial.

Without that, how to construct such a function, I'm a bit lost. I suppose if I can find a holomorphic function $f(z)$ whose real part full fill that it shall work, but playing around with the normal functions such as ${e}^{x}$, $\mathrm{sin}x$ etc, I can't find one with such property.

asked 2022-07-12

If argument of $\frac{z-{z}_{1}}{z-{z}_{2}}$ is $\frac{\pi}{4}$, find the locus of $z$.

${z}_{1}=2+3i$

${z}_{2}=6+9i$

Approach: I tried to solve the equation using diagram, basically plotting the points on the Argand plane. What I got is a circle with center $7+4i$ and a radius of $\sqrt{26}$ units. The two complex numbers given lie on this circle, and form a chord. Any point lying on the major arc of this chord satisfies the condition.

How exactly would I represent this as a locus of the point? And is there any other method that I can use that does not involve a diagram?

${z}_{1}=2+3i$

${z}_{2}=6+9i$

Approach: I tried to solve the equation using diagram, basically plotting the points on the Argand plane. What I got is a circle with center $7+4i$ and a radius of $\sqrt{26}$ units. The two complex numbers given lie on this circle, and form a chord. Any point lying on the major arc of this chord satisfies the condition.

How exactly would I represent this as a locus of the point? And is there any other method that I can use that does not involve a diagram?

asked 2022-08-17

Perhaps a rather elementary question, but I simply couldn't figure out the calculations on this one. Say one takes a circle centeblack at the origin with radius $R$. He or she then proceeds to place $N$ circles with radius $r$ ($R>r$) on the larger circles circumference equidistantly, so every $2\pi /N$ in the angular sense. What is then the relationship between $R$ and $r$ such that all neighboring circles exactly touch?

I've been trying to write down some equations with arc lengths and such for $N=4$, but I can't seem to get anything sensible out of it.

I've been trying to write down some equations with arc lengths and such for $N=4$, but I can't seem to get anything sensible out of it.

asked 2022-06-24

A circle has diameter AD of length 400.

B and C are points on the same arc of AD such that |AB|=|BC|=60.

What is the length |CD|?

B and C are points on the same arc of AD such that |AB|=|BC|=60.

What is the length |CD|?