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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.

50
votes
1answer
5k 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 ...
66
votes
6answers
43k 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 ...
51
votes
8answers
95k 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 ...
28
votes
4answers
4k 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 ...
55
votes
13answers
6k 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 ...
86
votes
18answers
23k 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 ...
9
votes
4answers
2k views

covariant and contravariant components and change of basis

I encountered the following in reading about covariant and contravariant: In those discussions, you may see words to the effect that covariant components transform in the same way as basis ...
3
votes
1answer
1k 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 ($\vec{e_1'},\vec{e_2'},\vec{...
3
votes
3answers
2k views

If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$?

Have $n$ vectors in $\mathbb{R}^n$. If the $n$ vectors are linearly independent, can we conclude that their span is $\mathbb{R}^n$?
2
votes
3answers
18k views

Find shortest distance between lines in 3D

Find shortest distance between lines given by $x-2/3=y-6/4=z+9/-4$ and $x+1/2=y+2/-6=z-3/1$ Is there any shortcut method for this problems?
12
votes
3answers
22k 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 ...
10
votes
1answer
6k 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\...
4
votes
3answers
4k views

Difference between orthogonal projection and least squares solution

When you find the least squares solution you solve $$A^TA = A\vec b$$ but to find the orthogonal projection into the "subspace" A, you multiply this result (the least squares solution) with the ...
2
votes
4answers
435 views

Why are the two dot product definitions equal?

I have an intuitive understanding of why $a\dot{}b=|a||b|\cos{\theta}$ geometrically. The projection of one vector onto another makes sense to me when explaining the origin of this geometric ...
55
votes
13answers
377k 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\...
31
votes
5answers
57k 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?
4
votes
2answers
4k views

Orthogonal Projection onto the $ {L}_{2} $ Unit Ball

On an article I'm reading, I find that: if $v$ is a vector, the projection of of $v$ on the unit ball is: $$p(v)=\frac{v}{\max\{1,\|v\|\}}$$ I know that a projection of a point $v$ into a space is the ...
3
votes
2answers
3k views

What is the intuitive way to understand Dot and Cross products of vectors?

Suppose, we are crossing a river with heavy current using a speed-boat. If $x$ is the vector for the current and $y$ is the vector for the speed-boat, then what do $x \cdot y$ and $x \times y$ ...
2
votes
1answer
301 views

Find the distance between two lines

\begin{equation} L_1:(x,y,z)=(-1,2,0)+t(0,-1,1) \\ L_2:(x,y,z)=(1,2,1)+s(1,-1,0) \end{equation} I have looked at multiple other questions but still can't solve it, the closest I got was this: ...
11
votes
2answers
31k 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 = \...
1
vote
2answers
2k views

Find plane by normal and instance point + distance between origin and plane

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
0
votes
2answers
556 views

Proof with 3D vectors

Let ${a} = \begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}$, ${b} = \begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}$, and ${c} = \begin{pmatrix}x_c\\y_c\\z_c\end{pmatrix}$. Show that $(x_a,y_a,z_a)$, $(x_b,y_b,...
25
votes
2answers
14k 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 ...
23
votes
5answers
45k 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?
5
votes
3answers
1k 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 ...
10
votes
2answers
6k views

Kronecker product and outer product confusion

I have two column vectors: \begin{equation} u = \left[\matrix{ 1 \cr 2\cr }\right] \end{equation} \begin{equation} v = \left[\matrix{ 4 \cr 4\cr }\right] \end{equation} I'm trying to ...
7
votes
4answers
1k views

determination of the volume of a parallelepiped

Here is a parallelepiped.I want to determine the volume of the parallelepiped. One of my friends said to me that the volume of the parallelepiped can be found out by the following formula. $${\...
4
votes
1answer
7k views

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the ...
3
votes
8answers
767 views

For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes?

While watching a video about dot products (https://www.youtube.com/watch?v=WDdR5s0C4cY), the following formula is presented for finding the angle between two vectors: For vectors $a$, and $b$, $$\...
4
votes
4answers
38k views

Find the equation of the plane knowing that it passes through 3 points

I have to find the equation of the plane that passes through $(0, 0, 0), (4, 0, -2), (0, 8, -6)$. I have done the following: The equation of the plane is of the form $$ax+by+cz+d=0$$ Since the ...
3
votes
1answer
1k views

polar coordinates vector equation of a rectangle

We can write the equation of the circle in vector form in polar coordinates as: $$\vec{r}=R\hat{r}$$ ; where 'R' is the radius of the circle. Similarly, can we write the vector equation for a ...
-1
votes
1answer
29 views

Expressing using properties of vectors [closed]

It is given in question that $x+a=(a.x)b$ and it is stated that a,b and x are vectors.It is also stated that $a.b$ not equal to $1$.The question ask to express $x$ in term $a$ and $b$,how do you this?
3
votes
1answer
255 views

Why do $n$ linearly independent vectors span $\mathbb{R}^{n}$?

Suppose we have $n$ linearly independent vectors $\mathbf{v}_{1}\ldots\mathbf{v}_{n}$ in $\mathbb{R}^{n}$. I know that they do span $\mathbb{R}^{n}$, because we can easily specify a non-singular map ...
1
vote
2answers
231 views

What are the “building blocks” of a vector?

Lets say I have a set of vectors $V$ that includes this vector: $$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$ I interpret it as $x = 1, y = 2, z = -1$ (that being three dimensions for this vector). I know ...
1
vote
1answer
81 views

Finding beam paths using reflection

I'm trying to figure this out on my own so no direct answers please - I really am looking for pointers on different ways to approach this problem. Given some dimensions, a point A, a point B, and a ...
1
vote
1answer
121 views

Question regarding basis vectors of root reference frame…

Probably my question is rather silly but then again I would rather ask you than going ahead and doing something even sillier. Right, in an old maths book(or at least what remains of it) I was ...
1
vote
2answers
1k views

Gradient steepest direction and normal to surface?

From this Maths SE question, I now understand the gradient to be the directional derivative that returns the steepest slope at a point. However, reading my textbooks, they all say that the gradient is ...
0
votes
2answers
119 views

Computing matrix-vector calculus derivatives

$x, a$ in $\mathbb R^n$, $A$ in $\mathbb R^{n\times n}$. Compute $d(x^T a)/dx$ and $d(x^T A x)/dx$. I'm not sure about how to think about these and how to do these. Can someone explain how to derive ...
24
votes
3answers
29k 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. Thanks in advance!
24
votes
8answers
7k 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 ...
15
votes
8answers
71k 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
5answers
6k 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 &= \...
6
votes
3answers
477 views

Coordinate free definition of $\nabla$ operator

There are a number of posts on this site asking similar questions and some of them have been answered (to my taste) at least partially but none give a complete answer that I am satisfied with. See ...
18
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 ...
8
votes
1answer
126 views

Counting vectors in $\mathbb{Z}_n^n$ with $0$ as a most common coordinate value

How can we count the number of vectors in $\mathbb{Z}_n^n$ that have $0$ as their strictly most common coordinate value appearing exactly $k$ times ? More precisely, if we denote by $$\alpha(\mathbf{...
5
votes
1answer
610 views

Can octonions be used to rotate 7-dimensional vectors?

A friend told me that, the same way you can represent a 3-vector as an imaginary quaternion then conjugate it by a unit quaternion with real part $\cos(\frac{\theta}{2})$ to rotate it by $\theta$ ...
5
votes
8answers
7k views

Scalar triple product - why equivalent to determinant?

I'm looking at the scalar triple product and I'm wondering: is there any demonstration (possibly a simple one) that $$ \mathbf{a} \cdot \left(\mathbf{b} \times \mathbf{c} \right)= \begin{bmatrix} ...
4
votes
8answers
1k views

Distance of two lines in $\mathbb{R}^3$

Using vector methods show that the distance between two non parallel lines $l_1$ and $l_2$ is given by $$d=\frac{|(\overrightarrow{v}_1 - \overrightarrow{v}_2) \cdot (\overrightarrow{ a}_1 \times \...
3
votes
5answers
626 views

Reasoning behind the cross products used to find area

Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross ...
9
votes
1answer
266 views

Why is $\arccos(-\frac 13)$ the optimal angle between bonds in a methane ($\rm CH_4$) molecule?

Background: In a CH4 molecule, there are 4 C-H bonds that repel each other. Essentially the mathematical problem is how to distribute 4 points on a unit sphere where the points have maximal mutual ...