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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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 ...
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 ...
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 ...
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 ...
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 ...
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\...
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 ...
5
votes
3answers
35k 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?
5
votes
3answers
8k 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 ...
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\...
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 ...
10
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
2k 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{...
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?
17
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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} ...
6
votes
4answers
946 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 ...
6
votes
2answers
716 views

Matrix Multiplication not associative when matrices are vectors?

Wikipedia states: Given three matrices A, B and C, the products (AB)C and A(BC) are defined if and only the number of columns of A equals the number of rows of B and the number of columns of B ...
4
votes
3answers
3k 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 $\mathbb{R}^n$?
3
votes
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 ...
1
vote
1answer
148 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 ...
34
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 ...
29
votes
5answers
63k 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?
11
votes
2answers
8k 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 ...
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 = \...
4
votes
2answers
5k 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 $x \times y$ ...
4
votes
2answers
3k views

**Location** of shortest distance between two skew lines in 3D?

I can find the shortest distance $d$ between two skew lines $\vec{V_1}$ and $\vec{V_2}$ in 3D space with $d=\left|\frac{(\vec{V_1}\times\vec{V_2})\cdot\vec{P_1P_2}}{|\vec{V_1}\times\vec{V_2}|}\right|$...
2
votes
1answer
371 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: ...
0
votes
2answers
736 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,...
8
votes
3answers
2k 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
1answer
4k views

Why is $\mathbb{R}^2$ not a subspace of $\mathbb{R}^3$?

I cannot understand why $\mathbb{R}^2$ is not a subspace of $\mathbb{R}^3$. My reasoning is as follows: Choose any elements $v_1$ and $v_2$ from $\mathbb{R}^2$, add them together you get an element ...
6
votes
8answers
8k 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} ...
8
votes
4answers
2k 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
9k 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 ...
4
votes
8answers
1k 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 $b$, $$\...
3
votes
2answers
2k 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 ...
4
votes
4answers
50k 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
2k 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 ...
0
votes
1answer
2k views

How to find the rotation matrix that will align an arbitrary vector to an axis

If I have a vector that starts at the origin, how can I find the transformation matrix that will align it with the positive y-axis. So it basically turns into a positive-y axis? EDIT: I also forgot ...
-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?
4
votes
3answers
13k views

What is the line of greatest slope on a plane?

Let $P$ be a plane in $\mathbb{R}^3$ that is inclined (neither horizontal nor vertical). When considering lines lying on $P$, it is sometimes said "$L$ is a line of greatest slope of $P$". What is ...
3
votes
2answers
2k 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 ...
3
votes
1answer
698 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 ...
2
votes
4answers
248 views

Quick doubt about multiplying vectors by scalars

I've just started studying linear algebra, and I'm having trouble understanding something that apparentely seems obvious to everyone else. My question is, why does multiplying a vector by a positive ...
1
vote
2answers
273 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
149 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 ...
0
votes
2answers
126 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 ...
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 ...
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. ...

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