# Tag Info

### 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....
• 3,223
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 $g$ ...
• 35.2k

### 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,...
• 97.9k
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 ...
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### 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 ...
• 3,223
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 ...
• 24k
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 $x, y \in ... • 11 3 votes ### Determining condition of coplanarity The idea is any two nonparallel vectors can span out a plane in$\mathbb{R}^3$. Let $$\textbf{u}:=5𝑎⃗+6𝑏⃗+7𝑐⃗,\textbf{v}:=7𝑎⃗+𝜆𝑏⃗+9𝑐⃗,\textbf{w}:=3𝑎⃗+20𝑏⃗+5𝑐⃗$$ Then the coplanar condition ... • 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 ... • 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+\... • 1,641 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 &...
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 $\angle KMX=90^\circ$,...