New answers tagged

2 votes

Find equation of plane using a system of equations

In $\mathbb R^3, A=(-2,3,4), u=(1,1,-1), l=A+\mathbb R u$ $B=(-2,0,2), v=(2,1,-1), g=B+\mathbb R v$ $\boxed{\color{blue}{\alpha : y+z=2}}$ The machine( geogebra ) uses another method : https://www....
Stéphane Jaouen's user avatar
2 votes
Accepted

Find equation of plane using a system of equations

The solution is $$y+z-\color{red}2=0,$$ not $y+z-7=0.$ Your point $Q\in l$ is not supposed to lie on the plane, which contains $g$ but is only parallel to $l$. You should take some point of $g$ ...
Anne Bauval's user avatar
  • 35.2k
2 votes

A Weak Type of Convexity for Smooth Jordan Domains in $\mathbb{R}^2$?

Definition. A subset $A$ is locally starlike if for every $a\in A$ there exists a neighborhood $U$ of $a$ in $A$ and a point $b\in int(U)$ such that $U$ is starlike with respect to $b$. More precisely,...
Moishe Kohan's user avatar
  • 97.9k
2 votes
Accepted

How to explicitely reduce the expression of the intersection of plan and sphere from 3D to 2D?

The radius of the circle is, as you found out, is $R = \dfrac{\sqrt{3}}{2} $ And the center of the circle is $ C = (0, 0, \dfrac{1}{2}) $ Two vectors of length equal to $R$, that are mutually ...
Hosam Hajeer's user avatar
  • 22.2k
0 votes

Distance of a point from a line measured parallel to a plane

The first approach is a good one. The problem in the second approach (I've checked all your calculations and get the same results ) has been pointed out by @Bob Dobbs: You reason as if you were in ...
Stéphane Jaouen's user avatar
2 votes
Accepted

Semantics of the angle between velocity vector and the positive $x$-axis

You almost said it yourself: "if the angle is constrained to be in the range...". Therefore, by contrapositive (or contradiction), it must be that the angle is not constrained to any bounded ...
Mark S.'s user avatar
  • 24k
0 votes

Determining condition of coplanarity

You can indeed view this problem as presenting 3 equations with 4 variables. However the assumption that you need as many equations as variables is false for several reasons. Example 1: Find $x, y \in ...
Vincent's user avatar
  • 11
3 votes

Determining condition of coplanarity

The idea is any two nonparallel vectors can span out a plane in $\mathbb{R}^3$. Let $$\textbf{u}:=5𝑎⃗+6𝑏⃗+7𝑐⃗,\textbf{v}:=7𝑎⃗+𝜆𝑏⃗+9𝑐⃗,\textbf{w}:=3𝑎⃗+20𝑏⃗+5𝑐⃗$$ Then the coplanar condition ...
Angae MT's user avatar
  • 1,031
2 votes

Determining condition of coplanarity

If they are coplanar then $\det\begin{bmatrix}5 &7 &3 \\6 &\lambda & 20 \\7 & 9 & 5\end{bmatrix}=0$, where the given matrix represents linear transformation in the basis ...
Widawensen's user avatar
  • 8,172
2 votes
Accepted

Determining condition of coplanarity

There's another approach which is imposing $$\mathbf u\cdot(\mathbf v\wedge\mathbf w)=0,$$ where $$\left\{\begin{align} &\mathbf u=5\mathbf a+6\mathbf b+7\mathbf c\\ &\mathbf v=7\mathbf a+\...
Joan S. Guillamet F.'s user avatar
1 vote

Determining condition of coplanarity

Even if you don't know about matrix associated with your system, You can try to translate them into manipulations on your system : $\begin{bmatrix}5 &7 &3 \\6 &\lambda & 20 \\7 &...
Stéphane Jaouen's user avatar
2 votes

An extension of Brahmagupta's theorem.

As K bisects AB and $\angle AMB=90^\circ$, $|MK|=|KB|$. $\angle MBK=\angle MCD$ by inscribed angles on AD. Therefore $\angle BMK=\angle MBK$. Let X be a point towards B such that $\angle KMX=90^\circ$,...
JMP's user avatar
  • 21.8k

Top 50 recent answers are included