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2
votes
2answers
43 views

Scalar triple product in terms of scalar products only

I am trying to express the scalar triple product $\bf a \cdot (b \times c)$ only in terms of scalar products $\bf a \cdot b$, $\bf b \cdot c$, $\bf c \cdot a$ and the lengths of vectors $a$, $b$ and ...
0
votes
0answers
15 views

From points on a latitude-longitude map to direction on hemisphere

I've wrote a program in matlab that following a given algorithm individuate some points on a latitude-longitude map. This is an example output: Now I need to map these points (in blue) to ...
0
votes
1answer
26 views

Three planar vectors $x,y,z$ such that $x$ is orthogonal to $y + z$ and $z$

Let $x$ be a non-zero vector, orthogonal to vectors $y + z$ and $z$, with $x, y, z \in \mathbb R^2$. Prove that $y$, $y - z$ and $z - y$ are orthogonal to $x$ and parallel to $z$. To prove they ...
1
vote
0answers
33 views

Intersection between 2 lines (3D). This doesn't have a solution does it?

so I was looking through an old exam and this question was given: The teachers answer was the point (9, -9, 21) I tried solving this myself, I got x = x, y = y, but I could not find a point where ...
0
votes
1answer
22 views

How to find an equation of a plane perpendicular to two other planes and passing through a point

Please, could anybody help me with the next problem. I have two planes: $$ 2x-y+5z+3=0 \ (\text{red plane})\\ x+3y-z-7=0 \ (\text{green plane}) $$ And I need to find a plane which is perpendicular ...
0
votes
4answers
104 views

To prove $\cos(A+B) = \cos A \cos B - \sin A \sin B$ [on hold]

How to prove the formula $\cos(A+B) = \cos A \cos B - \sin A \sin B $ by using cross product of two vectors?
-1
votes
0answers
23 views

Proving the formula $\cos(a+b)$ by cross product of vectors [on hold]

How can I prove the formula $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ using cross product of vectors?
0
votes
3answers
65 views

Does adding linearly independent vectors retain linear independence?

Suppose the vectors u, v, w are linearly independent and u'=u+v, v'=v+w and w'=u+w. I'd like to check if u', v', w' are also linearly independent. I know they can be linearly independent, such as if ...
2
votes
2answers
44 views

Referring to the unit normal vector

Is the unit normal vector or the normalized normal vector at a point on a surface the same as "the normal vector" at that same point (and surface)? Does saying, "the normal vector" imply it is ...
1
vote
2answers
23 views

Matricial differentiation $x x^{\top} b $

What is the drivative of $x x^{\top} b $ with respect to x, knowing that b is constant vector?
1
vote
1answer
54 views

Find an expression of the direction field

I have a directions vector field which I got empirically using quiver in Matlab. I want to find some analytical expressions that might work at least in part of the direction field. How can I ...
0
votes
1answer
15 views

How to find plane that's equidistant from the origin

Objective: Give the equation of a plane that crosses the axes at points equidistant from the origin. How do I make sure that the points $A(1,2,-2)$, $B(-5,1,1)$, $C(4,-3,1)$ are equidistant from the ...
1
vote
4answers
45 views

How do I find the perpendicular distance of a point from a plane?

Find the perpendicular distance of the point $(3,1,-2)$ from the plane $2x+y-2z = 8$. (The answer is $1$). I keep getting $3$ :( .. Thanks in advance.
0
votes
2answers
30 views

When are vectors $\vec a = (2, 3),\,\vec b = (x, 2)$ perpendicular/parallel

I have the following problem: Let $\vec a= (2, 3), \vec b = (x, 2)$ When $\vec a$ and $\vec b$ are perpendicular, $x=?$ When $\vec a$ and $\vec b$ are parallel, $x=?$ I have the key with the ...
2
votes
3answers
46 views

How to reverse matrix vector multiplication?

I'm using the simple matrix x vector multiplication below to calculate result. And now I wonder how can I calculate ...
-2
votes
1answer
35 views

Equation of a plane equidistant from 3 points

Question: Given 3 point (point A, point B, point C), find an equation to a plane that crosses the axes at points equidistant to the origin P[0,0,0]. Are the following steps the right way to approach ...
-4
votes
0answers
26 views

Division of elements of vectors with each other

Suppose we have a vector, like: x = [3,5,7,9,2,3] What does the division of elements from each other, left to right, indicates? Illustrating: ...
0
votes
1answer
23 views

How to know when a line is parallel to the xz-plane

What are some features of the equations of a line that is parallel to the xz plane, but does not lie on the plane, and is not parallel to any of the axes? So far all I got: -dot product of plane's ...
0
votes
2answers
20 views

Do you use the inner or outer degrees of a right-angled triangle when calculating the vertical component of a vector

I'm given the assignment of finding the vertical component of vector a - b. Below is an image of vector A. ||A|| = 6. I need to calculate side y. I was following the assignment and tried solving y ...
0
votes
1answer
6 views

Find the acute angle made by vector $OC$ and the x-axis.

Given that vector $OA$ = $3i+5j$, $OB$ = $-2i+6j$ and that $OC$ = $OA + OB$, calculate i) |OC|, ii) the acute angle made by vector $OC$ and the x-axis. I found i) $\sqrt122$ Please help me in ...
1
vote
2answers
34 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
1
vote
3answers
61 views

Find a basis of $A = (\{1, \sin(x), (\cos)^2(x), (\sin)^2(x)1\})$ and determining its dimension.

We consider a space F(R,R) of functions of R in R. Let A = ({1, \sin(x), $\cos^2(x)$, $\sin^2(x)$}) Find a basis of the vector subspace of F(R,R) and determine its dimension. So I used the identity ...
1
vote
0answers
18 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
0
votes
1answer
36 views

Integer Solutions to Cosine's Dot Product Formula

Say one wanted to test their students on the dot product formula without a calculator. One would (being a nice teacher and all) natural like to pick numbers in the plane that are "nice" and satisfy a ...
1
vote
2answers
27 views

Steps to creating 3 plane equations with 3 lines of intersection

I was wondering if anyone can give me pointers on to how to mathematically create 3 plane equations that meet in 3 lines. In other words, each plane intersects one another in a straight line (so it ...
0
votes
1answer
24 views

How to find normal of an intersecting plane

The equations of two different planes that intersect are the following: (1)---> x-3y+5z+8=0 (2)---> 5x+y-2z+7=0 It says that I must create a third plane that passes through the line of intersection ...
1
vote
2answers
65 views

What are the various branches of Vectors?

I just wanted to begin learning 'Vectors'. But I am completely confused where I would have to start! There is vector Algebra, Vector geometry, Vector Analysis and what not. So, I want to know what ...
0
votes
2answers
42 views

How to get a vector from its length and angle

How can I get the vector when I know its length and angle? E.g. length = 3 and angle = 40°.
0
votes
1answer
20 views

How to find angle between line and plane?

The normal of the plane is [-5,8,-14] and the direction vector of the line is [2,4,3]. I know that after using the equation cosθ=u⃗ ⋅v⃗ ||u⃗ ||⋅||v⃗ |, I must subtract 90º by the found angle in order ...
0
votes
0answers
42 views

Relation between componenet and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
0
votes
2answers
18 views

Position of a point with respect to two reference frames

I working on a project where doing some image processing detect objects using Kinect camera and then move it to a desired location with a help of robotic arm. In this project the sensor gives pixel ...
-2
votes
1answer
25 views

unit vector orthagonal to two vectors

How would I calculate the unit vectors here? What is the appropriate way to solve this? I do not completely understand which approach to take. Thanks!
0
votes
1answer
41 views

Vector calculation question…

in the formula for the calulation of the angle between 2 vectors $$\cos \theta \overset{\text{def}}= \dfrac{\vec\alpha \cdot \vec\beta}{|\vec \alpha|\cdot |\vec \beta|}$$ is the output angle is ...
0
votes
1answer
42 views

Partial derivative of a vector

I'm trying to show: $\displaystyle \frac{\partial} {\partial t}( \nabla(\phi))= \nabla\frac{\partial \phi} {\partial t} $ Am I allowed to write: $\displaystyle \frac{\partial} {\partial ...
1
vote
0answers
35 views

having trouble with a 3-dimensional basis-change problem/

Let $V$ be a 3d vector space with a chosen basis $\alpha=\{e_1,e_2,e_3\}, \beta=\{f_1,f_2,f_3\}$ for $V$ satisfying: $$\begin{align}e_1 & =f_1+f_2+f_3 \\ e_2 &=f_2+2f_3 \\ e_3 & =f_3 ...
1
vote
1answer
24 views

Potential function calc 3

I an unable to answer this question. I got a potential function of $V(x,y,z) =yzx^2$. I also know I must plug in $c(0)$ and $c(8)$. I got $c(0)=(0,0,1)$ and $c(8)=(64,0,e^{48})$. I then plug these ...
0
votes
0answers
25 views

Metric for movement in 2D space

I have a set of points that represent the coordinates of an object moving in the 2D space at different points in time. Using this points I want to get a "measurement" that will describe the general ...
2
votes
1answer
77 views

An inequality involving vectors

Let $n$ be a positive integer number. If $S$ is a finite set of vectors in the plane, let $N(S)$ denote the number of two-element subsets $\{\mathbf{v}, \mathbf{v'}\}$ of $S$ such that ...
0
votes
1answer
53 views

How to find the equation of two lines at a given point?

Here's the problem: Give the equations of two lines that meet at the point (-1, 5, 2) and which meet at right angles, but do not use that point in either of the equations. Any ideas?
1
vote
2answers
59 views

Vectors inside of a linear transformation.

so I am working on an algebra question and I need some help understanding what the question is asking. I found that the standard matrix of the linear transformation: $$T: \mathbb{R}^2 \to ...
3
votes
3answers
92 views

Why is the cross product of two vectors always orthogonal to the input vectors?

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors, we get the magnitude of the rotation or moment as $|\mathbf{a}||\mathbf{b}|\sin(\angle ab)$. Now, we are multiplying that with the unit vector ...
5
votes
3answers
326 views

Are vectors and covectors the same thing?

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) -- because the spaces overlap. However, a physicist ...
1
vote
1answer
10 views

Finding the conditions of (x,y,z,t) for them to belong to the span of a set of vectors

So I got this math exercise, and I don't know how to go about it: In $\mathbb{R}^4$, $S$ is the subspace spanned by the following set of vectors: $(1, 1, 1, 0) , (1, 2, 1, 1) , (2, 0, 1, 1) , (3, 0, ...
0
votes
1answer
34 views

How to write symmetric equation?

Usually, to write a symmetric equation, I know that we'd isolate the scalar multiple (also known as t) from the parametric equation so that each may equal to one another. However, after solving a ...
1
vote
1answer
26 views

Simple Trig Question / Introduction to Vectors Question

Sorry this is such a simple question; I'm just struggling a little with my trigonometry homework. An example question: "A ship sails due north (relative to the current) with a speed of 20 knots. The ...
0
votes
0answers
14 views

Properties of cross product ${\rm i}(a\times a^*)$

Given a complex 3-vector $a\in\mathbb{C}^3$, let $b$ be the following vector $$b={\rm i}(a\times a^*)$$ where $a^*$ is the element-wise complex conjugate of $a$. As can be easily shown by ...
2
votes
1answer
36 views

Where does the definition of the $L_0$ norm come from?

Where does the definition of the $L_0$ norm come from? $$\|x\|_0=|S|$$ Where $S=\{x_k:x_k\neq 0\}$
2
votes
0answers
68 views

How to find the “average” direction of a set of vectors?

I have a series of vector directions and I need to find the "average" direction. I am not looking for the overall direction which would be the sum of the directions and this can't be used in cases ...
0
votes
0answers
24 views

solve a system of equations with equal spans of vectors

If we know the matrices H1, H2, H3, H4, H5, H6, H7, H8, and H9 How can we solve the following system of equations: Span(H1 V1) = Span(H2 V2) = Span(H3 V3) Span(H4 V1) = Span(H5 V2) = Span(H6 V3) ...
0
votes
1answer
19 views

How to show a vector space is not closed under addition with elements not in the vector space.

Wasn't entirely sure how to word the title. What I'm trying to show is: Given $\vec{v}\in V$ and $\vec{w}\not\in V$, then $\vec{v}+\vec{w}\not\in V$ How would this statement be proven?