0
votes
0answers
3 views

Mirror a line over a plane

I am trying to mirror a line over a plane, but I am not sure if I am doing it right, so please tell me if something that I do is wrong. I have 2 points $A(1, 2, 1);B(-1,0,2)$ and I have to mirror the ...
0
votes
0answers
17 views

Finding the equation of a plane by using point-to-point distances

Assume that we have a plane $P_1$ whose equation is known. I need to find the equation of plane $P_2$. If we choose a point set $N = \{n_1, n_2, ...\}$ on $P_1$ and another point set $M = \{m_1, m_2, ...
0
votes
2answers
27 views

Find perpendicular vectors in subspace of $V_{3}$

Find all vectors of $V_{3}$ which are perpendicular to the vector $(7,0,-7)$ and belong to the subspace $L((0,-1,4), (6,-3,0)$. As a note, this is an extra question of a long exercise, the ...
0
votes
2answers
55 views

$\overline{x} \times \overline{a} = \overline{b}$ has a solution when $ \langle\overline{a},\overline{b} \rangle =0$

I'm trying to solve this exercise: Let $\overline{a} \neq \overline{0}$, $\overline{b}$ be two vectors of the Euclidean vector space $V_{3}$. Prove the equation $\overline{x} \times \overline{a} = ...
2
votes
2answers
65 views

check if point is on a plane (using Heron formula ?)

Is this true that if any of parameters a, b, c, d is equal to sum of three others then 4 points are on same plane? I am given 4 points in 3 dimensional space. Is this correct to state that all 4 ...
1
vote
2answers
36 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
1
vote
1answer
50 views

How to find out if four points are on the same plane, only by using distances?

There is a method called Cayley-Menger determinant in order to find if 3 points are collinear, 4 points are coplanar etc. provided that all the pairwise distances are given. However, in 2-D, there is ...
2
votes
1answer
58 views

Proof for $\cos(\alpha)^2 + \cos(\beta)^2 + \cos(\gamma)^2 = 1$ in Euclidean space

What is the proof for this formula: $$ \cos(\alpha)^2 + \cos(\beta)^2 + \cos(\gamma)^2 = 1, $$ where $\alpha$, $\beta$ and $\gamma$ are the angles between a vector and the base of a right-handed ...
2
votes
0answers
40 views

Cauchy-Schwarz on a Euclidian Space

I was thinking about this proof of the cauchy-schwarz inequality, I wanna show that $$|\langle u,v\rangle|\leq|u||v|$$. We know that, $$|\langle u,v\rangle| = ||u||v|\cos{\theta}|$$ where $\theta$ ...
1
vote
0answers
19 views

finding a locus in a $3$ dimensions

Given a Tetrahedron $OABC$ such that $O(0,0,0),A(a,0,0),B(0,b,0),C(0,0,c)$ ; $a,b,c$ are not zero. We build a plane $\pi$ that is parallel to $z$-axis and also to $AB$. Plane $\pi$ cuts plane $ABC$ ...
1
vote
0answers
102 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
0
votes
0answers
33 views

Quadric surfaces classification

I have an exam soon on quadrics. However, I don't have enough exercises, especially ones involving finding standard form of quadrics using affine and orthogonal transformations - rotation, translation ...
1
vote
2answers
144 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
4
votes
1answer
102 views

Coordinate Transformations

I am physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) between coordinate transformation of two kinds : Rotation of coordinate (and ...
1
vote
0answers
63 views

How to calculate center coordinates of two reverse arcs in 3D space

Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? ...
0
votes
1answer
34 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...
1
vote
2answers
54 views

Recognize conics from the standard equation

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if ...
1
vote
1answer
124 views

Line equation and distance between a line and a point in 3D space

I've looked for many similar questions but could not find an answer that helps. I have a point $p$ $(x,y,z)$ on a plane, and a normal to the plane $n$ $(a,b,c)$. I need to find the equation of a line ...
2
votes
1answer
46 views

When are $ r_1: ax + by + c=0 $ and $ r_2: a'x + b'y + c'=0$ distinct?

When are $ r_1: ax + by + c=0 $ and $ r_2: a'x + b'y + c'=0$ distinct? I think, if $$rank\left(\begin{array}{cc} a & b \\ a' & b' \end{array} \right)\neq 1$$ then $r_1$ and $r_2$ are ...
5
votes
1answer
85 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
1
vote
1answer
39 views

Proving $proj_{proj_{\vec u} \vec v} \vec v=proj_{\vec u} \vec v$

Can anyone show me how to prove: $proj_{proj_{\vec u} \vec v} \vec v=proj_{\vec u} \vec v$? I got confused trying to prove it (not geometrically)... Thanks in advance!
2
votes
1answer
118 views

Geometric Interpretation of Jacobi identity for cross product

Is there a geometric "reason" for the Jacobi identity for cross products? Some geometric equality of some area ...? All proofs I know work by some form of linear algebra (or use the interpretation as ...
0
votes
2answers
136 views

affine transformation of triangle

$(a,b,c),(a',b',c') \in (\mathbb{R}^2)^3$ triples of pw different points which are not colinear. Show: There's an affine transformation $x \mapsto Ax+s$ with $Aa+s=a',Ab+s=b',Ac+s=c'$ mapping one ...
1
vote
2answers
474 views

Rotation in 3D (coordinate system transformation)

How do I rotate a point around point [0,0,0] in 3D. In picture I draw specific situation for illustration. At first I know point G[x,y,z] and I will tranfer it on axiz Z, where distance to center is ...
0
votes
1answer
31 views

When is this line part of this plane..

given is: $ g: \vec x = \frac{1}{15} \begin{pmatrix} 14 \\ 13 \\ 7 \\ \end{pmatrix} + r* \begin{pmatrix}1\\0\\2 \end{pmatrix}$ and $ E: ax_1 * 2x_2 + x_3 = 4 $ What is "a", when $ g \in E $ ? ...
1
vote
1answer
49 views

Vector Projection with respect to another vector

I have learnt about orthogonal projections, but now there is a new problem regarding non orthogonal projections. As seen in the image, given vector d, i would like to project vector v to the line with ...
0
votes
1answer
102 views

Direction cosines

I have two vectors, A and B that meet at point R. The vector RA has a magnitude 2km and direction cosines cos alpha=0.768, cos beta=0.384, cos gamma=0.512. The vector RB has a magnitude 4km and ...
1
vote
1answer
249 views

How to find the angle between vectors which is not necessarily the smallest angle

I understand that in order to calculate the angle between two vectors one does the arccos of the results of the dot product divided by the product of the magnitude of the two vectors. However, this ...
5
votes
2answers
168 views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
4
votes
4answers
219 views

Geometric visualization of covector?

How could I geometrically visualize a linear functional?
5
votes
3answers
655 views

What is the difference between vector components and its coordinates?

Some mathematitians told me that vector components and coordinates are different things. They say that vector $F^n$ always has N components but coordinates depend on chosen basis and, therefore, it is ...
2
votes
1answer
89 views

Find the distance between two lines in $ \Bbb R^3 $ [duplicate]

There are two lines in $ \Bbb R^3 $ given in parametric form: $$ l_1: \left\{ \begin{aligned} x &= x_1 +a_1t\\ y &= y_1 +b_1t \\ z &= z_1 +c_1t \\ \end{aligned} \right. $$ $$ l_2: ...
1
vote
3answers
52 views

Convert a line in $ \Bbb R^3 $ given as intersection of two planes to parametric form.

We have a line in $ \Bbb R^3 $ given as intersetion of two planes: $$ \left\{ \begin{aligned} A_1x+B_1y+C_1z + D_1 &=0 \\ A_2x+B_2y+C_2z + D_2 &=0 \\ \end{aligned} \right. $$ How to ...
5
votes
2answers
78 views

How does this vector addition work in geometry?

I saw the accepted answer to the question: Finding a point along a line a certain distance away from another point! I am not getting how to use it actually to find the coordinates of the new point at ...
8
votes
5answers
196 views

How to think of $\vec{u}-\vec{v}$

Assume I have two vectors, $\vec{u}$ and $\vec{v}$. I know that I can think of their sum via Triangle or Parallelogram Law, but I'm having trouble knowing which way the vector would point depending on ...
1
vote
1answer
57 views

Find the transform

I have the paper with 3 points on it. I have also a photo of this paper. How can I determine where is the paper on the photo, if I know just the positions of these points? And are 3 points enough? It ...
0
votes
0answers
77 views

Derivation of $Ax+By+C=0$

Below I've tried to derive the standard form of the linear equation. Please check my derivation. Object: Derive $Ax+By+C=0$. The slope-intercept form of the linear equation can be given as: ...
1
vote
2answers
171 views

Max and min value of $7x+8y$ in a given half-plane limited by straight lines?

So, there are four inequalities: $$\begin{eqnarray*} y &\geq &-3x+15; \\ y &\leq &-11/3x+56/3; \\ x &\geq &0; \\ y &\geq &0. \end{eqnarray*}$$ If we draw all those ...
2
votes
1answer
177 views

Foundations of analytic geometry

I was just wondering about the formal foundations of analytic geometry, I mean axiomatically. I've noticed along my course of linear algebra that the axioms of vectorial space already include the fact ...
0
votes
2answers
158 views

Non-degenerate quadratic form and non-singular matrix

Let $(V,Q)$ be a finite-dimensional quadratic space over a field $\mathbb{K}$. From my definition, $Q$ is non-degenerate if $\operatorname{rad}(V)=\{0\}$. How can I prove that $Q$ is non-degenerate ...
1
vote
1answer
46 views

Unit vectors orthogonal to L

I have a line $L$ in $\mathbb{R}^2$ that passes through two points: $u = [9;7]$ $v = [1;-5]$ How do I find all unit vectors orthogonal to $L$? I know: $[x;y] * [8;12] = 0$ and $x^2 + y^2 = 1$ ...
2
votes
3answers
825 views

Find all unit vectors orthogonal to line with two given points

I have a line $L$ in $\mathbb{R}^2$ that passes through two points: $[9;7]$ and $[1;-5]$ How do I find all unit vectors orthogonal to $L$?
2
votes
3answers
272 views

Simultaneous Equations and Vectors

The question I am currently working on is, "...find $a$ and $b$ such that $\vec{v} = a \vec{u} + b \vec{w}$, where $ \vec{u} = \langle 1,2 \rangle$, $\vec{w} = \langle 1,-1 \rangle$, and $\vec{v} = ...
0
votes
2answers
30 views

Computation with scalar product

Let $\vec{a}$ and $\vec{b}$ be vectors from $V_3$. Suppose, that $|\vec{a}| = 1$, $|\vec{b}|=2$ and the angle between $\vec{a},\vec{b}$ is $\frac{\pi}{3}$. Use the properties of scalar product and ...
2
votes
4answers
3k views

Find intersection of two 3D lines

I have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values. Please some body tell me how can I find the intersection of these lines. EDIT: By using ...
2
votes
3answers
114 views

What kind of software can be used to solves systems of equations?

For example, I have to solve the following equations: $$\left\{\begin{align*} &x^2 + y^2 + z^2 = 1\\ &Ax + By + Cz = 0 \end{align*}\right.$$ for $y$ and $z$, where $x$, $A$, $B$ and $C$ are ...
1
vote
1answer
162 views

Finding two points that have a defined distance between two intersecting lines

I was given a test yesterday, a test which unfortunately I was unable to study to. In it was a question that was too hard that it became our homework for the whole week. It says that on lines ...
1
vote
1answer
75 views

Locus perpendicular to a plane in $\mathcal{R}^4$

I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me.. In $\mathcal{R}^4$ I have a plane $\pi$ and a point P. I have to find the ...
4
votes
3answers
778 views

Find a plane perpendicular to a plane passing by point

In $\mathbb R^4$ I have: $$\pi: \begin{cases} x+y-z+q+1=0 \\ 2x+3y+z-3q=0\end{cases}$$ I have to find $\pi' \bot$ $ \pi $ and passing by $P=(0,1,0,1)$. How can I do that? Thanks a lot!
1
vote
1answer
591 views

Calculating an average plane from a series of points using Numpy [closed]

Given N, 3D position vectors, I want to find the best-fit plane of those vectors. I found a suggested answer here that said: Calculate the average of the points (easy) Calculate the Covariance ...