Math Is Fun Forum

A 5-sided polygon $ABCDE$ is inscribed in a circle centered at the origin. Define the lines
\begin{align*}
\ell_{ABC} &= \text{Line through the centroid of $\triangle ABC$ perpendicular to $\overline{DE}$},\\
\ell_{BCD} &= \text{Line through the centroid of $\triangle BCD$ perpendicular to $\overline{AE}$}, \\
\ell_{CDE} &= \text{Line through the centroid of $\triangle CDE$ perpendicular to $\overline{AB}$}, \\
\ell_{DEA} &= \text{Line through the centroid of $\triangle DEA$ perpendicular to $\overline{BC}$}, \\
\ell_{EAB} &= \text{Line through the centroid of $\triangle EAB$  perpendicular to $\overline{CD}$}. \\
\end{align*}As we see, these are lines going through the centroid of a triangle formed by three consecutive vertices, perpendicular to the line segment formed by the other two vertices.
Prove that $\ell_{ABC}, \ell_{BCD}, \ell_{CDE},\ell_{DEA}$ and $\ell_{EAB}$ are concurrent, and find the expression for the position vector of the point they all go through.

This problem involves vectors in ways I don't understand. I have no idea how to go about this, can you walk me through how to solve it? Thanks!

Let $A, B$ and $C$ be points in $3$-space:
[asy]
size(200);
import three;

currentprojection = orthographic(3, 1, 0.5);
real big = 6;
triple O = (0,0,0);
triple X = (1, 0, 0);
triple Y = (0, 1, 0);
triple Z = (0, 0, 1);
triple A = (0, 2, -1);
triple B = (2, 0, 3);
triple C = (3, -1, 5);

draw(O--big*X, Arrow3(size = 0.2cm));
draw(O-- -big*X, Arrow3(size = 0.2cm));
draw(O-- big*Y, Arrow3(size = 0.2cm));
draw(O-- -big*Y, Arrow3(size = 0.2cm));
draw(O-- big*Z, Arrow3(size = 0.2cm));
draw(O-- -big*Z, Arrow3(size = 0.2cm));

label("$x$", big*X, S);
label("$y$", big*Y, S);
label("$z$", big*Z, W);

dot("$A$", A, S);
dot("$B$", B, S);
dot("$C$", C, S);
[/asy]Prove that if $A, B$ and $C$ are collinear, then
\[\overrightarrow{A}\times \overrightarrow{B} + \overrightarrow{B}\times \overrightarrow{C} + \overrightarrow{C}\times \overrightarrow{A} =\begin{pmatrix} 0 \\ 0\\ 0 \end{pmatrix}.\]

I'm stuck, can anyone explain how to solve this?

(a) Let

and let

Show that

sends each of

and

to scalar multiples of themselves, and find the value of these scalars.

(b) Let

be a positive integer. Use part (a) to find the vectors

(c) Write the vector

as a linear combination of

and

.

(d) Using parts (a), (b), and (c), calculate

I need to write a detailed solution for all four of these; if you could walk me through that, that would be great. Thanks!

We're going to consider the matrix

(a) Let

.

Find the

such that

(b) Find a formula for

.

(You don't need to prove your answer, but explain how you found it.)

(c) Using parts (a) and (b), find a formula for

I haven't been able to get anywhere on this problem; an explained solution would be much appreciated. Thanks!

(a) Show that any two-dimensional vector can be expressed in the form

where

and

are real numbers.

(b) Let

and

be non-zero vectors. Show that any two-dimensional vector can be expressed in the form:

where

and

are real numbers, if and only if of the vectors

and

, one vector is not a scalar multiple of the other vector.

I have no idea how to start; can you work me through how to prove it?

Find all real solutions for

I could use detailed help and/or a solution, the sooner the better.
The only work I have is simplifying and proving that 2^x - 1 and x have the same sign.
Thanks!
!nval!d_us3rnam3