Questions tagged [vectors]

Use this tag for questions and problems involving vectors, e.g., in an Euclidean plane or space. More abstract questions, might better be tagged vector-spaces, linear-algebra, etc.

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124
votes
18answers
50k views

What is the difference between a point and a vector?

I understand that a vector has direction and magnitude whereas a point doesn't. However, in the course notes that I am using, it is stated that a point is the same as a vector. Also, can you do ...
112
votes
12answers
12k views

Is there an “inverted” dot product?

The dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: $$\mathbf{a} \cdot \mathbf{b} =\sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$$ What about the quantity? $$\...
90
votes
18answers
28k views

Is arrow notation for vectors “not mathematically mature”?

Assuming that we can't bold our variables (say, we're writing math as opposed to typing it), is it "not mathematically mature" to put an arrow over a vector? I ask this because in my linear algebra ...
81
votes
6answers
51k views

Is the vector cross product only defined for 3D?

Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as $$ \vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal ...
69
votes
9answers
126k views

What does the dot product of two vectors represent?

I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, $2$ and $3$, we say that it ...
60
votes
15answers
414k views

How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\...
57
votes
1answer
7k views

What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
55
votes
13answers
7k views

Why is cross product defined in the way that it is?

$\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector? Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a ...
43
votes
5answers
74k views

Adding two polar vectors

Is there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?
33
votes
4answers
292k views

Determine if vectors are linearly independent

Determine if the following set of vectors is linearly independent: $$\left[\begin{array}{r}2\\2\\0\end{array}\right],\left[\begin{array}{r}1\\-1\\1\end{array}\right],\left[\begin{array}{r}4\\2\\-2\...
33
votes
2answers
20k views

Row vector vs. Column vector

I'm a student in an elementary linear algebra course. Without bashing on my professor, I must say that s/he is very poor at answering questions, often not addressing the question itself. Throughout ...
31
votes
4answers
8k views

Cross product in higher dimensions

Suppose we have a vector $(a,b)$ in $2$-space. Then the vector $(-b,a)$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear. Suppose instead we have ...
30
votes
3answers
43k views

Difference between sum and direct sum

What is the difference between sum of two vectors and direct sum of two vector subspaces? My textbook is confusing about it. Any help would be appreciated.
28
votes
8answers
13k views

What does the symbol nabla indicate?

First up, this question differs from the other ones on this site as I would like to know the isolated meaning of nabla if that makes sense. Meanwhile, other questions might ask what it means in ...
28
votes
5answers
62k views

What is vector division?

My question is: We have addition, subtraction and muliplication of vectors. Why cannot we define vector division? What is division of vectors?
23
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11answers
5k views

What equation produces this curve?

I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple - a gentle curve which begins and ...
23
votes
2answers
22k views

how does the dot product determine similarity?

I want to know how the dot product can determine whether two vectors are similar? I know that the formula $$\cos(\theta) = \frac{u \cdot v }{ ||u||\,||v||}$$ means something, but don't know what.
20
votes
1answer
9k views

Why is the SVM margin equal to $\frac{2}{\|\mathbf{w}\|}$?

I am reading the Wikipedia article about Support Vector Machine and I don't understand how they compute the distance between two hyperplanes. In the article, By using geometry, we find the ...
19
votes
8answers
3k views

3rd type of vector multiplication beside dot/cross product?

I was reading up on how to find the square root of i , and I learned that multiplication of complex numbers could be viewed geometrically by viewing the complex numbers as coordinates on the complex ...
18
votes
4answers
7k views

Show that $(2,0,4) , (4,1,-1) , (6,7,7)$ form a right triangle

What I tried: Let $A(2,0,4)$, $B(4,1,-1)$, $C(6,7,7)$ then $$\vec{AB}=(2,1,-5), \vec{AC}=(4,7,3), \vec{BC}=(2,6,8)$$ Then I calculated the angle between vectors: $$\begin{aligned} \alpha_1 &= \...
18
votes
4answers
32k views

Why is the derivative of a vector orthogonal to the vector itself?

$R(t) \cdot R'(t) = 0$, which is what every source I can find tells me. Even though I understand the proof I don't understand the underlying concept. If $R(t)\cdot R'(t) = 0$, then $R'(t)$ is ...
18
votes
2answers
1k views

Is there any distinction between these products: scalar, dot, inner?

I hope you will forgive a math question that comes up in physics contexts where language is loose. This question migrated from Physics SE. I'm finding that I sometimes don't know what kind of product ...
18
votes
1answer
763 views

Why do we treat differentials as infinitesimals, even when it's not rigorous

From single-variable calculus where we first encounter differentials we are told fairly often that differentials are not to be treated as infinitesimal quantities/objects (but we are never really told ...
17
votes
8answers
90k views

How to find the distance between two planes?

The following show you the whole question. Find the distance d bewteen two planes \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray} ...
17
votes
2answers
50k views

What does $\|u\|$ mean?

What does $\left\Vert \mathbf{u}\right\Vert$ mean in this equation? How would this equation be performed? I'm extremely terrible in discrete mathematics and a simplistic answer would be ideal. (Don't ...
17
votes
4answers
1k views

When is a vector “glued” to the origin?

Let $V$ be a real finite-dimensional vector space (I guess this forces $V$ to be $\mathbb{R}^n$). My intuition is that a vector $v\in V$ must be "glued" to the origin, since the origin is the only ...
16
votes
3answers
8k views

Average norm of a N-dimensional vector given by a normal distribution

I'm interested in knowing what is the expected value of the norm of a vector obtained from a gaussian distribution in function of the number of dimensions $N$ and $\sigma$, i.e: $$E[\|x\|_2],\quad x\...
16
votes
1answer
10k views

In the context of vectors, is there a difference between the terms “magnitude” and “length”?

I noticed vectors are usually said to have "length" and "direction", but then it is said that people want to find the "magnitude". Is this just a difference in terminology or is there something more ...
15
votes
4answers
44k views

What does this (double absolute value like) notation mean?

Here, $$\left\lVert\frac{\partial\bf x}{\partial s}\times\frac{\partial\bf x}{\partial t}\right\rVert$$ the inside will at last be a vector. and two absolute value signs have covered it. what does ...
15
votes
3answers
31k views

Euclidean distance and dot product

I've been reading that the Euclidean distance between two points, and the dot product of the two points, are related. Specifically, the Euclidean distance is equal to the square root of the dot ...
15
votes
3answers
17k views

Standard notation for sum of vector elements?

I can think of multiple ways of writing the sum of a vector $\mathbf{v}$'s elements, but is there a standard operator for this? Using "programming" notation it is typically sum($v$), but this seems ...
15
votes
2answers
53k views

Formula to project a vector onto a plane

I have a reference plane formed by 3 points in $\mathbb{R}^3$ – $A, B$ and $C$. I have a 4th point, $D$. I would like to project the vector $\vec{BD}$ onto the reference plane as well as project ...
14
votes
10answers
7k views

Is there a math function to find an element in a vector?

I would like to write mathematically, if possible, the following statement: Given a vector $x=[1,4,5,3]$ and an integer $j=3$, find the position of $j$ in $x$? How to write this mathematically? ...
14
votes
3answers
2k views

Does cross product have an identity?

Does cross product have an identity? I.e. Does there exist some $\vec{id}\in \mathbb{R}^3$ such that $$\vec{id} \times \vec{v} = \vec{v}\times \vec{id} = \vec{v} $$ for all $\vec{v}\in \mathbb{R}^...
14
votes
2answers
17k views

Proving that the dot product is distributive?

I know that one can prove that the dot product, as defined "algebraically", is distributive. However, to show the algebraic formula for the dot product, one needs to use the distributive property in ...
14
votes
1answer
3k views

What is the difference between vector-valued functions and parametric equations?

So as it is, I'm now starting to cover vector-valued functions in my Calculus III class. While studying the topic, I noticed that it seemed to be the exact same thing as parametric equations. I know ...
14
votes
4answers
280 views

Reflect a ray off a circle so it hits another point

my problem is the following: I have two points ($e$ and $p$) in a 2D space and I am trying to figure out where on the circle is the reflection of $p$ as seen from $e$. $$$$ $$$$ So the way I ...
13
votes
3answers
21k views

How to sort vertices of a polygon in counter clockwise order?

How to sort vertices of a polygon in counter clockwise order? I want to create a function (algorithm) which compares two vectors $\vec v$ and $\vec u$ which are vertices in a polygon. It should ...
12
votes
4answers
3k views

English names for vector beginning and end

I've done some research, but since English is not my native language, I'm struggling to find an answer to this: Given a vector, what do you call its beginning and end points? The best I've found so ...
12
votes
4answers
962 views

Is there a possible geometric method to find length of this equilateral triangle?

Problem Given that $AD \parallel BC$, $|AB| = |AD|$, $\angle A=120^{\circ}$, $E$ is the midpoint of $AD$, point $F$ lies on $BD$, $\triangle EFC$ is a equilateral triangle and $|AB|=4$, find the ...
12
votes
3answers
19k views

How do I find a dual basis given the following basis?

$V = \Bbb{R}^3$ and has basis $\mathcal{B} = \{\vec{e_1}-\vec{e_2},\vec{e_1}+\vec{e_2},\vec{e_3}\}$ How do I find the dual basis? This is not homework, but an example that I am struggling to grasp. ...
12
votes
4answers
1k views

A vector is defined to have a magnitude and *a* direction, but the zero vector has no *single* direction. So, how is the zero vector a vector?

The popular definition of a vector is A vector is an object that has both a magnitude and a direction. We know that zero vector has no specific single direction. Then how can it be a vector?
12
votes
3answers
450 views

Is there a name for matrix product with reversed indices?

The typical matrix product is as follows: $$ (\mathbf{A}\mathbf{B})_{ij} = \sum_{k=1}^m A_{ik}B_{kj}\,. $$ Is there a name or characterization for one such as $$(\mathbf{A}\mathbf{B})_{ij} = \sum_{...
12
votes
3answers
1k views

Why doesn't the dot product give you the coefficients of the linear combination?

So the setting is $\Bbb R^{2}$. Let's pick two unit vectors that are linearly independent. Say: $v_{1}= \begin{bmatrix} \frac{1}{2} \\ \frac{\sqrt{3}}{2}\end{bmatrix}$ and $v_{2} = \begin{bmatrix} ...
12
votes
2answers
41k views

Do four dimensional vectors have a cross product property? [duplicate]

We know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$ Where : $ \vec A = (A_i; A_j; A_k)$ $ \vec B = (B_i; B_j; B_k)$ $ \vec C = (C_i; C_j; C_k)$ $C_i = \...
12
votes
7answers
270 views

Crux problem #33 with vector approach

On the sides $CA$ and $CB$ of an isosceles right-angled triangle $ABC$, points $D$ and $E$ are chosen such that $|CD|=|CE|$. The perpendiculars from $D$ and $C$ on $AE$ intersect the hypotenuse $AB$ ...
11
votes
4answers
7k views

Is zero a scalar?

Is zero considered a scalar? In other words, is $\begin{bmatrix}0\\0\\\end{bmatrix}$ a scalar multiple of $\begin{bmatrix}a\\b\\\end{bmatrix}$ where $a$ and $b$ are real numbers?
11
votes
2answers
9k views

Set, n-Tuple, Vector and Matrix — links and differences

I know this question has been asked like 1000 times, however all supplied answers were not really satisfying to me. My question concerns the similarities and differences between these mathematical ...
11
votes
4answers
1k views

What is the intuition behind the definition of the differential of a function?

What is the intuition behind the definition of a differential of a function in differential geometry? i.e. $$df(p)(v_{p}) =v_{p} (f)(p) $$ where $v_{p} \in T_{p} M$ is a vector in the tangent space to ...
11
votes
4answers
2k views

The difference between applying a rotation matrix to a vector (points) and to a matrix (transformation)

Suppose that the rotation matrix is defined as $\mathbf{R}$. Then in order to rotate a vector and a matrix, the following expressions are, respectively, used $\mathbf{u'}=\mathbf{R} \mathbf{u}$ and ...

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