Use this tag for questions involving vectors, quantities that have magnitude and direction.

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0answers
15 views

Prove the lines are concurrent (using vectors)

Problem: Let $A$, $B$, $C$, $D$, and $E$ be points on a circle. For any three points, we draw the line going through the centroid of the triangle formed by these three points that is perpendicular to ...
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1answer
12 views

Differentiating a function that includes vectors using the chain rule

I am trying to differentiate the function: $$g(x) = f(3\vec k + x(\vec l + \vec k))$$ where $\vec k$ and $\vec l$ are in $\mathbb R^n$ and $x$ is in $\mathbb R$. I think I need to use the chain ...
1
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1answer
25 views

If $v\not=0$, then $v/\|v\|$ has norm 1

The question is: Show that if $\vec v$ is a non-zero vector in $\mathbb R^n$ then $\left( \dfrac{1}{||\vec v||} \right ) \vec v$ has norm $1$. I assume that $\vec v=(v_1,v_2,v_3,...,v_n)$ , ...
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2answers
15 views

line integral problem with a circle

I want to evaluate the line integral $$ \int_C (2xy^3+\cos x)\,dx + (3x^2y^2+5x)\,dy $$ where $C$ is the circle $x^2+y^2=64$ I parametrized the circle by $r(t)=8\cos t \ \hat{i} +8\sin t \ \hat{j}$ ...
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1answer
38 views

Integrand for a set of points

I need help finding what I should be integrating when the question asks to find the double integral to find the volume of the tetrahedron given the points $(0,0,0),(3,0,0),(2,1,0),(3,0,4)$. Would the ...
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0answers
21 views

If the Unit Vectors are equal, Are the directions equal too?

Given that the unit vector of x = unit vector of y, can we conclude that the Direction (or Sign) of x is always equal to Direction of y?
2
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1answer
33 views

Interpreting 3D parametric equations

I've been working through a problem and I have managed to reduce it to the following:$$x=\frac{2r}{3}\cos\theta - \frac{r}{3}\sin\theta$$ $$y=\frac{2r}{3}\sin\theta - \frac{r}{3}\cos\theta$$ $$z = ...
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0answers
30 views

Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($\vec{e_1},\vec{e_2},\vec{e_3}$) and ...
1
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2answers
23 views

Is there solution for equation which is recursive?

I have this following vector equation:$$\vec x = \vec x_0 + (1/2) * f(\vec x, t)t^2$$ where $\vec x$ has initial value when first fed into function $f$. All vectors are 3 dimensional and function $f$ ...
1
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1answer
13 views

Linear Combination of Vectors, given their norms and angles.

If I have $3$ vectors, $u,v,w$, and $|u|=6,|v|=9,|w|=12$, how do I write w as a linear combination of u and v? (Just in case the diagram isn't clear, the angle between vector u and v is 30 degrees and ...
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0answers
7 views

Find coordinates of point that satisfy given conditions

I have A(1,2,3) , B(-1,0,1), C(1,-1,1) which are points in $\mathbb{R}^3$. I'm trying to find another point H such that AH${\parallel}$AC and BH${\perp}$AC. I set H = ($h_1$, $h_2$, $h_3$), and took ...
1
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1answer
13 views

Rotate a vector into a plane spanned by two other vectors

In an application test that I had to do for a job recently, I was asked the following question (I quote): “Given three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. Compute the rotation ...
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1answer
21 views

Use the orthonormality of $u,v,w$ to write the following vectors as linear combinations of $u,v$ and $w$

Let $V$ be the vector space $\mathbb R^3$ with inner product $$(v,w)=3(v_1w_1)-2(v_1w_2)-2(v_2w_1)+5(v_2w_2)-3(v_2w_3)-3(v_3w_2)+3(v_3w_3)$$ where $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$. Part 1 ...
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1answer
25 views

Prove that the vectors u,v,w are orthonormal in V

Let V be the vector space R3 with inner product (v,w)=3(v1w1)-2(v1w2)-2(v2w1)+5(v2w2)-3(v2w3)-3(v3w2)+3(v3w3) where v=v1,v2,v3 and w=w1,w2,w3 Prove that the vectors u=(1,1,1), ...
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1answer
10 views

How to find coordinates of point on intersect two lines

Let $A(a1,a2,a3),B(b1,b2,b3),C(c1,c2,c3),D(d1,d2,d3)$ be four different points. Let $E(e1,e2,e3)$ be the point on intersect lines $AB$ and $CD$. How to find $e1,e2,e3$? I tried this way: vector ...
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3answers
19 views

Let V be an inner product space. Prove that for any 2 vectors, (u,v)=1/4(|u+v|^2-|u-v|^2)

Let V be an inner product space. Prove that for any 2 vectors (u,v)=1/4(|u+v|^2-|u-v|^2) Thanks very much for any help
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1answer
28 views

Proof of Vector Space Axioms [on hold]

Where can I find detailed proof of vector space axioms? Any reference to a book, website or video lecture.
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2answers
19 views

Computing orthogonal projection

The question asks: A vector u and a line L in R^2 are given, compute the orthogonal projection w of u on L. u=[3,4] and y=-x In one example i was given two ...
0
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1answer
13 views

Maximizing vector scalar product

Suppose we have vectors as $$\vec{V} = 2\hat{i} + \hat{j} -\hat{k} $$ and $$\vec{W} = \hat{i} + 3\hat{k} $$ So what will maximum value of $$k=[ \vec{U} \vec{V} \vec{W}]$$ for some unit vector ...
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0answers
18 views

How does one come up with the vector product

In every math book I had so far, I just read how the vector product (cross product) can be calculated and some of the books showed a way to proof it. However, I am still asking myself how someone came ...
0
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2answers
23 views

Find the equation of the plane that contains:

Find an equation for the plane containing the lines $$x = 5y = \frac{z + 1}{4}$$ and $$\begin{cases} x = t \\ y = 2t\\ z = 6t − 1 \end{cases}.$$ I know that finding two points will allow me to find ...
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0answers
11 views

Orthonormal basis and representation of a vector with angle

Given two orthonormal vectors $u_1$ and $u_2$ which form a basis for $\mathbb{R}^2$. The vector $u \in \mathbb{R}^2$ can be represent as $$u = \cos \theta \ u_1 + \sin \theta \ u_2$$ where $\theta$ is ...
2
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1answer
17 views

Split vector by zeros

I have got a problem with splitting a vector by zeros. I have a vector for example $$v=[1\ 3\ 2\ 6\ 4\ 0\ 0\ 2\ 4\ 6\ 0\ 0\ 0\ 3\ 1]$$ I need to get vectors like $$v_1=[1\ 3\ 2\ 6\ 4]$$ $$v_2=[2\ ...
2
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0answers
14 views

How to express vectors with more than 2 components in complex coordinates

It is straightforward to extend the notion of a 2D vector in the Cartesian x,y plane to 3D (x,y,z) or to any D. Sometimes it is useful to express vectors in the complex plane, where the 2D vector has ...
1
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2answers
23 views

Finding the point of incidence of a light ray on a plane

I have that a ray of light is emitted from the point $(3,-2,-1)$ and reflected off the plane $x-2y-2z=0$. The reflected ray passes through the point $(4,-1,-6)$. I have to find the point at which the ...
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0answers
30 views

Question on distance between 2 lines and distance from origin to plane

So I have this vector equation which I can't solve. The question is in the attached image. I'm having trouble doing part a(ii) and b(i) and b(ii). With regards to part a(ii) here's what I know so ...
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1answer
20 views

How does cos theta equals to dot product of OP and OQ/ [mod(OP)*mod(OQ)]

O is the origin OP and OQ are vectors, for simplicity lets denote P as vector with magnitude 1 facing angle of 45 degrees from positive x axis. Let's denote Q as vector with magnitude 2 facing ...
0
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1answer
13 views

Given normal to plane and a point in it, find a unit vector in plane

I have a plane for which the unit normal vector and a point in the plane are known. I want to find a unit vector lying in the plane (any one). While there are dirty ways of doing this, I remember from ...
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2answers
30 views

Vector notation for “not including” index

I was wondering how to write vector notation with an index which is not included in the vector. In sets we can write, $$ A=\{0,1,2,3,4\},$$ then if we don't want to include the element $\{0\}$ we ...
2
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1answer
44 views

Finding a line through 4 other lines!

This one's probably easy, but I'm dreadfully stuck and can't seem to figure out a decent method. I have the following lines: $$a: \vec{x}(\lambda)= \left( \begin{array}{ccc} 4 \\ -2 \\ -2 ...
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0answers
23 views

Vector expression in terms of other vectors.

Question : Vectors $\vec x ,\vec y$ and $\vec z$ each of magnitude $\sqrt2$ , make angle of 60° with each other, if : $\vec x\times (\vec y\times \vec z)=\vec a$ , $\vec y\times (\vec z \times \vec ...
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0answers
24 views

Find boundaries of a shape

I have a bunch of points separated one from each other, that form a random shape (as shown on the picture) For each of the points there is known x,y coordinates on the vector. What I'm trying to do ...
2
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2answers
22 views

Calculating clockwise/anti-clockwise angles from a point

I'm currently trying to work out if an angle is a clockwise or anti-clockwise rotation about a point. I used the equation: a.b = ||a|| ||b|| cos(A) to calculate the angle between the two vectors in 3D ...
1
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1answer
19 views

Why is the graph of this vector equation a semi circle?

I have: $\mathbf{r}(t) = (t^2 - 1)\mathbf{j} + 2t\mathbf{k}$, $-1\leq t \leq 1$ and $\mathbf{r}(t) = (2\cos t)\mathbf{i} + 2(\sin t)\mathbf{k}$, $0 \leq t \leq \pi$. I've already graphed them but I ...
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0answers
25 views

Is there a formula for the direction of a line, not slope of a line?

This is for a CAD program I am currently developing. I trace a line that goes forward/backward and an arc that follows that goes left or right. I am aware that positive slope means the line goes ...
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2answers
22 views

Find a vector orthogonal to both $u $and $j+k$

So, $u=i-3j+2k$ I understand how to solve these type of problems with two given vectors, however I am lost on this. How do I find a vector orthogonal to both $u$ and $j+k$?
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0answers
20 views

Unique representation of each point in 3d space by Linear combination of 3 mutually perpendicular vectors.

I intuitively accepted that there is an unique representation of any point in a 3d space by linear combination of 3 mutually perp. vectors. But now I'm wondering is this an axiom or a theorem? If ...
0
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1answer
26 views

Flatten 3D VectorA so it's perpendicular to VectorB

Basically I have 2 3D vectors: Vector A (green) and vector B(red). I need to calculate a third vector that is perpendicular to VectorA (green) but points in the same direction than VectorB (red). ...
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1answer
4 views

Find Z Rotation based on X and Y Vector

I have a vector $(x,y) = (x_2 - x_1, y_2 - y_1)$. I have an arrow pointing to 0 degrees. With vector $(x, y)$, how can I find the number of degrees (0 - 360) that will be the direction the arrow ...
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2answers
11 views

Deriving a perpendicular vector to a plane from two parallel vectors

Given a point on a plane and two vectors that are parallel to that plane how can we derive a vector that is perpendicular to that plane? I am trying to find the equation of a plane and I need this ...
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2answers
26 views

Vector equation with cross product and unit vector

Does anybody know how to solve the equation $\mathbf{a} + \mathbf{b} \times \hat{\mathbf{v}} = c \hat{\mathbf{v}},$ where $\mathbf{a}$ and $\mathbf{b}$ are given real vectors, for the unit vector ...
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0answers
19 views

Intersection of lines in 3-D space.

I have a line $l$ that passes through the point $(3,2,-1)$ and am told that this line intersects the two lines with parametric representations: $(x,y,z)=(1+t, t, -5+t)$ and $(x,y,z)=(10+5t, 5+t, ...
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1answer
34 views

Multivariate Analysis

I have the following exercise: Given: $f(x)= < x,w>$ show that $\nabla f(x)=w$ and given $f(x)=\|Bx-c\|^2_2$ show that $\nabla f(x)=2B^T(Bx-c)$, where $f:\mathbb{R}^d \rightarrow \mathbb{R}, w ...
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0answers
10 views

Express the $i^{\text{th}}$-component of $y:=-(D+L)^{-1}Ux$ for lower and upper triangular matrices $L$ and $U$ and a diagonal matrix $D$

Let $A\in\mathbb{R}^{n\times n}$ be decomposed into $$A=L+D+U$$ where $L$ and $U$ are strictly lower and upper triangular and $D$ be diagonal. Now, consider$^1$ $$B:=-(D+L)^{-1}U$$ and let ...
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0answers
8 views

Solution to a particular quadratic vector equation

I have to solve for $y$ from a quadratic vector equation of the form $$y^{T}Py + 2q^{T}y = 0,$$ where $P \in \mathbf{R}^{n \times n}$ is positive semidefinite, and $q \in \mathbf{R}^{n}$. I got some ...
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0answers
30 views

Aerial Camera Ground Footprint Calculation

I have a very simple math problem, but I cannot seem to figure it out. I need to calculate what portion of the ground will be visible when viewed from a UAV mounted camera. I believe I have it solved ...
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0answers
44 views

Definition of a position vector

If the point $A$ has position vector $\mathbf a$ and the point $B$ has position vector $\mathbf b$, is the vector $\mathbf b-a$ a position vector? It is tied to one location.
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1answer
279 views

Tangent, Normal, Binormal Vectors, Curvature and Torsion

a) Find the unit tangent, normal, binormal vectors T N B and the curvature and torsion at a general point on the following curves; r = $t$i + $\frac{t^2}{2}$j + $\frac{t^3}{3}$k, ($0 \le t \le ...
2
votes
1answer
22 views

Vectors inner product on $\mathbb R$

Let $u_1,...,u_k$ vectors in $\mathbb R ^ n$ such that for every i,j we have $u_i \cdot u_j < 0$, and I want to show that $k\leq n+1$. I tried somehow use the fact that each vector seperates the ...
0
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1answer
29 views

How do you find the parametric and cartesian equation and include three points a, b and c?

Given three points $a = (1,4,0), b=(2,1,5), c=(3,5,2)$ in $\mathbb{R^3}$, find each of the following $iii)$ A parametric equation of the plane containing $a$, $b$ and $c$ My solution: ...