[Community Question] Geometry: Neep help justifying a vector relation given in a quation

One of our user asked:

i'm trying to do a question for which I was given the following line equations:

$\underline r = \underline a + \lambda \underline u$

$\underline r' = \underline a' + \lambda' \underline u'$

They then gave me this relationship without any justification, i've been trying to get my head around it but have not had much luck.

$\lvert \underline r-\underline r'\rvert^2\lvert \underline u \times \underline u'\rvert^2=\lvert (\underline a - \underline a') \cdot (\underline u \times \underline u')\lvert^2+\lvert (\underline r - \underline r') \times (\underline u \times \underline u')\lvert^2$

I know that

$\lvert (\underline r - \underline r') \times (\underline u \times \underline u')\lvert = \lvert \underline r-\underline r'\rvert\lvert \underline u \times \underline u'\rvert\sin \theta $

but cant get any further.

Comments

[Community Question] Linear-algebra: non-negative matrix satisfying two conditions

One of our user asked: A real matrix $B$ is called non-negative if every entry is non-negative. We will denote this by $B\ge 0$ . I want to find a non-negative matrix $B$ satisfying the following two conditions: (1) $(I-B)^{-1}$ exists but not non-negative. Here $I$ is the identity matrix. (2) There is a non-zero and non-negative vector $\vec{d}$ such that $(I-B)^{-1}\vec{d}\ge 0$ . I tried all the $2\times 2$ matrices, but it did not work. I conjecture that such a $B$ does not exist, but don't know how to prove it.

Calculus: Intersection of hyperplanes in $\mathbb{R}^5$

Consider the $4$ hyperplanes in $\mathbb{R}^5$ given by the equations $$x_1-x_2+x_3+x_4+2x_5=0$$ $$2x_1-x_2+6x_3+2x_4+6x_5=0$$ $$3x_1-3x_2+3x_3+4x_4+7x_5=0$$ $$x_1+x_2+9x_3+x_4+6x_5=0$$ Let $V \leq \mathbb{R}^5$ be the intersection of these hyperplanes. Find a basis for $V$ .

[Community Question] Geometry: The limit about the line connecting the intersection of a circle and the $y$-axis and the intersection of the shrinking circle and a fixed circle

One of our user asked: There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$ , $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$ -axis. What happens to $R$ as $C_2$ shrinks, that is, as $r \to 0^+$ ? (The figure is made with GeoGebra ) In order to solve this problem, I made a script using GeoGebra in which the circle $C_2$ is a dynamic one whose radius $r$ can be adjusted with a slider. As I set $r \to 0^+$ , the figure seems to suggest that $R \to (4,0)$ . In particular, this is the state with $r = 0.001$ , in which $R$ is reported to be $(3.9999997523053,0)$ : However, I would like to find out a way to prove (or disprove, though unlikely) my guess that $$\lim_{r \to 0} R = (4,0).$$ But I have little idea. Any help would be appreciated.

Win At Mathematics

Search This Blog