2
votes
1answer
48 views

Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
-1
votes
2answers
38 views

Determine the matrix of the reflections in the fol­lowing plane in $\Bbb R^3$. [closed]

$2x_1-2x_2-x_3= 0$ How would I go about approaching this problem?
1
vote
3answers
67 views

Curiosity about Kronecker's Delta?

My professor gave this subject in the class (analytic geometry) and I thought it was very complicated, then I just decided to open Wikipedia entry on Kronecker's Delta and discovered it is quite ...
0
votes
1answer
26 views

Line parallel to plane

See if the line e is parallel with to the plane $α$. If not, find the intersecting point. $$\begin{align} α: & \quad \quad x-3y+z+1=0 \\ e: & \quad \{x+y-z=3, 2x-y-4z=3\} \end{align}$$
1
vote
0answers
22 views

How can I find the volume of this prism and points B, C, D and F?

In the triangular prism, A = (0, -1, 1), E = (0, -3, 0). C and D belong to line s: x - 1 = y = y - z. How can I find the prism's volume and the coordinates of B, C, D and F points?
2
votes
3answers
40 views

Understanding vector projection

I'm learning about vector projection. I understand how to perform it, but I still can't understand what it actually means and what it gives me. Here is a common definition: Vector projection of a ...
2
votes
3answers
172 views

Distance of a point to a line

Find the distance from the point $S(2,2,1)$ to the line $x=2+t,y=2+t,z=2+t$. How can I find the distance of a point in $3D$ to a line?
0
votes
4answers
277 views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
0
votes
1answer
27 views

To find an intersection point between two planes with only the direction vector

Find the intersection between two planes $x−3y−2z = 2$ and $2x+y+3z = 1$ Solution: $(1)$$\quad n_1 \times n_2 =\langle −7,7,7\rangle =7 \langle −1,1,1\rangle$. $(2)$ To find one intersection ...
0
votes
1answer
16 views

About a/the definition of plane.

Let $P$ be a point in 3-space and consider a located vector $ \overrightarrow {0N}$. We define the plane passing through $P$ perpendicular to $ \overrightarrow {0N}$ to be the collection of all ...
2
votes
2answers
31 views

Prove $(|OP|)+ |PQ|)^2 > |OQ|^2$

I did all the algebra and for some reason I'm getting 0 > $y_2^2$ which is clearly wrong. Where did I mess up at?
0
votes
1answer
59 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
0
votes
1answer
12 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 ...
2
votes
1answer
117 views

Finding the equation of a plane in 3-D by using point-to-point distances

Assume that we have a plane $P(a,b,c,d)$ whose equation is unknown. We know that there is a point set $N = \{n_1, n_2, ...\}$ and $\forall n_i \in N$, $n_i$ is on $P$. Also, $\forall n_i, n_j \in N$, ...
0
votes
2answers
32 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
63 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
99 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
41 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
210 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
71 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
44 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
21 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
142 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
44 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
189 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 ...
5
votes
1answer
128 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
84 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
41 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
61 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
206 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
47 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
86 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
40 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
156 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
145 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
574 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
54 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
115 views

Direction cosines

I have two vectors, $\vec A$ and $\vec B$ that meet at point R. The vector $\vec {RA}$ has a magnitude 2km and direction cosines $\cos(\alpha)$=0.768, $\cos(\beta)$=0.384, $\cos(\gamma)$=0.512. The ...
1
vote
1answer
260 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
196 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
243 views

Geometric visualization of covector?

How could I geometrically visualize a linear functional?
5
votes
3answers
803 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
100 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
82 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
204 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
61 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 ...
1
vote
2answers
212 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
211 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 ...