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 ...
1
vote
2answers
28 views

Writing a vector as the sum of two other vectors.

Suppose you have 2 vectors $\vec a = (1,1,2)$ and $\vec b = (3,4,-2)$, how would you write $\vec a$ as the sum of 2 vectors $\vec c$ and $\vec d$ where $\vec c$ is in the direction of $\vec b$ and ...
0
votes
1answer
46 views

Image and Kernel of different linear maps and their dimension

I'm trying to determine the image and the kernel of different linear maps. I understood well the theory but I can not transfer the knowledge of the books I have read to specific linear maps. 1) ...
3
votes
0answers
19 views

Higher Dimensional Right-Hand Rule

In seven dimensions, the cross product makes sense. Without resorting to nonvector tensors or exterior products (although they can be used to further explain), how does one perform this cross product ...
0
votes
2answers
33 views

How to find a normal vector from an equation in the form f(x,y)?

If I have an equation $f(x,y)$ which given the $x$ and $y$ coordinate, it gives you the $z$ coordinate. How can I find the normal (directional) vector of the the point $(x,y,f(x,y))$? This would be ...
0
votes
0answers
9 views

Cross Product of Covectors

Is the vector/cross product defined for covectors (in the dual space) or is it, strictly speaking, only defined for vectors themselves? I would imagine that it works fine for covectors but I wanted to ...
10
votes
2answers
817 views

Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
0
votes
1answer
57 views

Prove that $g(t) \times \frac{d}{dt} g(t) = 0$

If the vectorial function $r = g(t)$, with values in $\mathbb R^3$ and where $t\in\mathbb R$, is a solution of the differential equation $\frac{d^2}{dt^2} r(t) = t^2 r(t)$, such that $g(0) = 0$, ...
0
votes
1answer
48 views

Find the length and direction of $u \times v$ and $v \times u$

So I was given two vectors: $u=-8i- 2j- 4k$, and $v=2i+2j+k$. I was able to figure out the cross product of $u\times v$ which is $6i-12k$, and $v \times u$ which is $-6i+12k$. However, I need help ...
1
vote
2answers
317 views

Equation of plane that goes for intersection of 2 planes and is perpindicular to another plane

Really don't know what to do here, went to a tutor neither did he. Okay first the problem: Find the equation of the plane that passes through the line of intersection of the planes x − z = 2 and y + ...
2
votes
2answers
151 views

How to find all 3 orthogonal vectors to a 4D vector

For a program I'm writing, I need to find the vectors orthogonal to a given vector rotated at an arbitrary angle, and in 4D. It is a unit vector. For 3D, I found the two orthogonal vectors like ...
5
votes
0answers
42 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
2
votes
1answer
55 views

Why and how are quaternions 'bilinear'?

What does it mean when we say that quaternion composition is 'bilinear'? I have observed that some authors write quaternion multiplication as: While others specify: Excuse the poor images, ...
2
votes
1answer
42 views

If $U \times V$ is a unit vector, $|U| = \sqrt{3}/3, |V| = 2$ and $U\cdot V > 0$, then what is the angle between $U$ and $V$?

If $U \times V$ is a unit vector, $|U| = \sqrt{3}/3$, $|V| = 2$ and $U \cdot V > 0$, then what is the angle between $U$ and $V$? Is this just dot product? $U \cdot V = |U| |V| \cos(\theta)$? Solve ...
0
votes
2answers
69 views

Computing cross product using norm and angle

Sorry for the weird title, if someone finds a better title for my problem be my guest to edit it ;) For $\mathbf{v,w} $ in R³ with $\mathbf{||v||=1 ;||w||=4; \theta =\frac{2\pi}{3}}$ Solve ...
1
vote
1answer
45 views

Find vectors vertical to given vectors with certain length

Given the vectors $\mathbf{u,v}$ in R³, determine all vectors that are vertical to $\mathbf{u}$ and $\mathbf{v}$ with length = 1 Every vector $\mathbf{x'}$ that is to be found must meet these ...
1
vote
3answers
48 views

Reordering vector product

If I have vectors $a, b, c \in \mathbb{R}^3$, and if we have e.g. $a = b\times c$, is there any way to express $b$ in terms of the other two?
1
vote
1answer
97 views

Solution set to cross product

If $\vec a,\vec b \in \mathbb{R}^3$ with $|\vec a|\ne0$ show that the equation $\vec a \times \vec u =\vec b$ has a solution if and only if $a \cdot b = 0$ and find all the solutions in this case. ...
-3
votes
2answers
149 views

Why cross product's formulas defined in this way? [closed]

Why cross product's formulas defined in this way? When mathematicians need to define cross product?
1
vote
1answer
159 views

Test of handedness

I'm reading a book on linear algebra, where the author gives a method to test the handedness or chirality of a given set of 3 basis vectors. if (v1 x v2) . v3 > 0 then it's right-handed, while if ...
1
vote
2answers
189 views

Vectors and Cross Product

I have these two questions regarding the Cross Product. 1.) You are looking down at a map. A vector $u$ with $|u| = 3$ points north and a vector $v$ with $|v| = 10$ points northeast. What is $|u ...
1
vote
1answer
361 views

How to generate an ordered list of vertices of a cube from a face and a normal vector

Consider a cube with faces we'll call "left", "right", "front", "back", "top" and "bottom". The cube can be described by $0 \le x,y,z \le 1$. To name the faces, we'll say $x$ extends to the right, ...
-1
votes
2answers
691 views

Find the sine sign given a pair of 3D vectors

I want to find the exact sine between two vectors in 3-dimensional space. Data: $x$: vector $y$: vector $z = \Vert x \times y \Vert$ I have tried this: $$\sin \alpha = \frac{\Vert z\Vert}{( ...
19
votes
6answers
4k 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 ...
2
votes
1answer
111 views

Vector question, solving $r\wedge a=b$ and $r\wedge c=d$, with conditions

I am stuck on the following Show that the vector equation $r\wedge a=b$ has a solution $$r=\lambda a + \frac {a \wedge b}{|a|^{2}}$$ Show that the vector $r\wedge a=b$ and $r\wedge c=d$, with ...
2
votes
2answers
156 views

Vectors question

I'm trying to prove whether the followings statements are true or not. I would appreciate your help, as I'm not sure how to begin. Given: $ u,x_n \in \mathbb{R}^3$ and for every $n$, let $x_{n+1}=u ...
5
votes
1answer
756 views

Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but ...