For questions on cross products.

learn more… | top users | synonyms

0
votes
1answer
33 views

A proof to a vector identity

I do not know how to prove this, can anybody help me out with that? Consider five vectors: $\vec{a},\vec{b},\vec{c}, \vec{p}, \vec{q} \in \mathbb{R}^3$ then: ...
3
votes
4answers
69 views

How does computing the determinant of a matrix with unit vectors give you the Cross Product?

Say you had $(a_x,a_y,a_z)\times(b_x,b_y,b_z)$, you would set up a matrix like the following: And the resulting would be your Cross Product or the coordinates of an orthogonal vector. My question ...
0
votes
2answers
36 views

Can the cross product of two non-invertible matrices be invertible?

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible? Just used to help understand a Linear Transformation assignment question, don't ...
1
vote
2answers
66 views

How can we determine if two vectors are parallel?

What are the minimal number of products like dot cross that can give us information if two vectors are parallel ? What can we say if V*W = 1 assuming V and W are not unit vectors.
0
votes
2answers
49 views

How to calculate one of the vectors that generate a given cross-product?

Given the vector: $$\vec b=(-0.361728, 0.116631, 0.924960)$$ and it's cross-product: $$\vec a \times \vec b=(-0.877913, 0.291252, -0.380054)$$ How do I calculate $\vec a$ ? It's been a while since ...
0
votes
1answer
48 views

Lack of associativity of cross product vs associativity of the exterior product

Can someone remind me in a nutshell why the associativity of the exterior product fails to transfer to the cross product? (It's been over a decade since I had to deal with the former back in school.) ...
2
votes
1answer
37 views

Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
0
votes
0answers
24 views

Homology of $T^3$ generated by three copies of $T^2$

Why is the second homology of $T^3=S^1\times S^1\times S^1$ generated by $S^1\times S^1\times \{\mathrm{pt}.\}$, $S^1\times \{\mathrm{pt}.\}\times S^1$ and $ \{\mathrm{pt}.\}\times S^1\times S^1$? ...
0
votes
0answers
20 views

Decompose cyclic sum of crossproducts into two cyclic sums?

Suppose you have $6$ points $a_i\in\mathbb{R}^3$ $i\in\{1,..,6\}$ such that all triangles with vertices $0, a_i, a_{i+1}$ for $i\in\{1,..,5\}$ do not degenerate (I dont know if this assumption is ...
0
votes
4answers
38 views

Finding the exact value of the sine of the angle between a line and a plane

I have done part (a). For part (b), I know the principle of how to do it, I tried to use the cross product to find the exact value of the sine of the angle. So I found PQ which is ...
1
vote
2answers
43 views

Volume of tetrahedron using cross and dot product

Consider the tetrahedron in the image. Prove that the value of the tetrahedron is given by $\frac16 |a \times b| \cdot c$ So far, what I did was I know volume of the tetrahedron is equal to the ...
6
votes
2answers
379 views

Proof for vectors involving cross and dot product

Prove that for any two vectors $\mathbf a$ and $\mathbf b$, $\lvert \mathbf a \times \mathbf b \rvert^2 + (\mathbf a \cdot \mathbf b)^2 = \lvert \mathbf a \rvert^2 \, \lvert \mathbf b \rvert^2$. ...
0
votes
1answer
33 views

Is this a correct identity for the Kronecker delta and the Alternating Tensor?

If $\varepsilon_{ijk}$ is the alternating tensor and $\delta_{in}$ is the Kronecker delta, am I correct in thinking that $$ \delta_{in}\varepsilon_{ijk} = \varepsilon_{ink} $$ If not, what is the ...
9
votes
2answers
103 views

On defining cross (vector) product.

This has been bugging me for years so I finally decided to "derive" (for lack of a better term) the definition of the cross product in $\mathbb R{^3}$. Here was my method for finding a vector: ...
6
votes
1answer
72 views

Slick proof of cross product identities

The cross product between vectors in $\mathbb{R}^3$ obeys two pleasant identities (sometimes named after Lagrange), namely $a\times(b\times c)=b(a\cdot c)-c(a\cdot b)$ $(a\times b)\cdot(c\times ...
0
votes
0answers
23 views

Construct 3D plane from 2 points and minimize angle of two vectors with its normal

I have as input two points $P, Q \in E^3$ and two vectors $\vec{v}_1, \vec{v}_2 \in R^3$. I need to construct a plane $(\vec{n}, d)$ such that the two points are in the plane and the angles between ...
1
vote
2answers
35 views

How do I evaluate the following vector product

I need to evaluate the following $$((\vec{a}\times\vec{b})\times\vec{a})_i((\vec{a}\times\vec{b})\times\vec{a})_j.$$ I am assuming Levi civita notation would be useful, but couldn't utilise it. Does ...
1
vote
0answers
9 views

Discrete cross-correlation: How to do cross correlation in FFT with segmented windows?

I have a discrete signal: $x=[i_1, i_2, ..., i_n]$ I would like to do the cross correlation between sub-groups of (k number of) consecutive number of $x$, for example: $xcorr([i_1, i_2, ..., i_k], ...
0
votes
1answer
26 views

cross product of E and H

Can anyone explain how they came up with the product of E and H ? I don't understand why the exponent of E cross H are multiplied by 2. Thanks
0
votes
0answers
26 views

Using the Jacobian matrix to find surface area without a change of basis.

http://mathinsight.org/parametrized_surface_area_examples In reading through the example in the above link, it's straightforward to find the surface area for a cone as follows. Find the surface ...
0
votes
0answers
20 views

Norm of cross product

We define cross product $a\times b$ as a determinant of the following matrix: $$\begin{pmatrix} i&j&k\\ a_1&a_2&a_3\\ b_1&b_2&b_3 \end{pmatrix} $$ where $i,j,k$ forms the ...
0
votes
1answer
95 views

Show matrix transformation, finding kernel & range of it

Let $a=(a1,a2,a3)$ be a fixed vector in $\Bbb R^3$. Define the cross product $a \times v$ of $a$ and another vector $v=(v_1,v_2,v_3) \in \Bbb R^3$ as $$a \times v = \det\begin{bmatrix} e_1 ...
-1
votes
1answer
63 views

Relationship between cartesian product and cross product?

Is there any relation between cartesian product and cross product? Or is it just the same symbol?
0
votes
0answers
33 views

Angle using cross product

I have a situation. Please refer to a figure below: I have r1, r2, Ɵ1, Ɵ2 as well the reference line. I want to find out angle Φ(phi). i.e. (angle PBA). Edit_1 The link provided solves the ...
0
votes
1answer
25 views

The Normal Vector of two 3D vector's Cross Product [duplicate]

I am wondering why the cross product of two vectors in $\mathbb{R}^3$ would get the Normal Vector of the plane generated by them? I know it is the definition, but I am still wondering why we can get ...
0
votes
0answers
29 views

Problem with triple cross product proof

When trying to prove bac-cab rule, I get to point, where I don´t know, what is true. I have $$ (\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})a_jb_lc_m=a_jb_ic_j-a_lb_lc_i $$ but when $$ ...
0
votes
0answers
21 views

Proof of the reconcilation of the geometric form of cross product with the algebraic form.

In Arfken's "Mathematical Methods for physicists" he stated that: $(A\times B)\cdot(A\times B) = A^2B^2-(A\cdot B)^2=A^2B^2-A^2B^2\cos^2(\theta)$ How did he arrived to that? He said that he is ...
0
votes
2answers
44 views

Proving bac-cab $A \times (B \times C)=(A\cdot C)B-(A\cdot B)C$

I was asked to prove :$A \times (B \times C)=(A\cdot C)B-(A\cdot B)C$ using vector multiplication of $3$ dimension I chose $A=(a_x,a_y,a_z)$, $B=(b_x,b_y,b_z)$, $C=(c_x,c_y,c_z)$ and start with the ...
0
votes
1answer
65 views

The vector quadruple product - Index notation

I am looking to prove the following using index notation, but I am getting stuck on obtaining the RHS of the statement. Thus, I wanted to be 100% sure (by asking you guys) that the RHS of the ...
1
vote
1answer
45 views

How does crossing two 3d vectors produce a third one that is perpendicular to both? [duplicate]

Can someone help me understand the cross product a little better, for me it makes more sense for the new vector to be somewhere between the original vectors and closer to the bigger one but that would ...
1
vote
1answer
33 views

Cross-products of groups and their normal subgroups

If there exist groups $A$ and $B$, as well as their respective normal subgroups $C$ and $D$, then it is possible to prove that $(C\times D) \triangleleft (A\times B)$. I, however, have no idea where ...
0
votes
0answers
33 views

Theorems for Perpendicular and Parallel Vectors

I know the following about lines: If two lines are parallel to a third line, then they are parallel to each other. If two lines are perpendicular to a third line, then they are parallel to each ...
0
votes
0answers
35 views

Uniqueness of cross product

My professor in E&M told us that the cross product is the only possible product of vectors that produces another vector. I didn't ask him what he meant by that because he is Russian and can't ...
0
votes
0answers
52 views

Vector, C, orthogonal to both vectors A and B (which are orthogonal to each other)

Vectors A and B are orthogonal. To then find a vector C which is orthogonal to both A and B do I use the triple product (A x B = C) or do I use the dot product again and set equal to zero?
0
votes
1answer
73 views

Finding the Angle Between Two Vectors Using Cosine Law

When given any two vectors, it is possible to use the dot product to find the angle between the vectors without having any ambiguity as it uses the inverse cosine function. However, when we use the ...
0
votes
1answer
28 views

finding line in plane perpendicular to another line

I have a vector that I need to find. It is in the (111) plane. The vector I need to find is perpendicular to another vector which is parallel to [01-1]. To be clear: I want to find A which lies in ...
1
vote
2answers
26 views

Order of evaluation for the vector cross product

Since the vector cross product is non-associative, is the is the expression A x B x C (without parentheses) meaningless? Or is there a convention for the order of evaluation (left to right or right ...
0
votes
2answers
54 views

Why does the magnitude of the cross-product of a and b give the area of a parallelogram spanned by a and b?

I tried looking it up but many websites just state it without proof and without intuition. I'm hoping to learn it a little bit better so that I don't forget how to compute the Jacobian when working ...
1
vote
0answers
31 views

units in math, cross product

The cross product of two vectors has length equal to the area of the parallelogram they generate. The cross product is also a vector and thus has dimensions. But the units of those dimensions are ...
2
votes
1answer
43 views

A high-level reason that $(a \times b) \cdot ((b \times c) \times (c \times a)) = (a \cdot (b \times c))^2$

I can do the algebra to prove this identity: $$(\mathbf{a} \times \mathbf{b}) \cdot ((\mathbf{b} \times \mathbf{c}) \times (\mathbf{c} \times \mathbf{a})) = (\mathbf{a} \cdot (\mathbf{b} \times ...
1
vote
0answers
15 views

Compute $f\nabla\times\mathbf{r}$

I am trying to compute $f\nabla\times\mathbf{r}$ for $$ f(x,y,z) = \left(x^2+y^2+z^2\right)^{-\frac{3}{2}} \\ \mathbf{r} = x\mathbf{i}+y\mathbf{j}+z\mathbf{k} $$ I have computed $\nabla ...
3
votes
4answers
45 views

A high-level reason that $u \cdot (v \times w) = (u \times v) \cdot w$?

I can do the algebra to show that for $u, v, w \in \mathbb{R}^3$, this identity is true: $$u \cdot (v \times w) = (u \times v) \cdot w$$ But is there a more high-level reason? I didn't expect the ...
3
votes
2answers
53 views

Vector calculus problem

I have to solve this: $$[(\nabla \times \nabla)\cdot \nabla](x^2 + y^2 + z^2)$$ But I am really drowning in the sand.. Can anybody help me please?
0
votes
2answers
39 views

Trying to show $|\overrightarrow{a}\times\overrightarrow{b}|^2=|\overrightarrow{a}|^2|\overrightarrow{b}|^2-(\overrightarrow{a}⋅\overrightarrow{b})^2$

If $\overrightarrow{a} = \langle a_1, a_2, a_3 \rangle$ and $\overrightarrow{b} = \langle b_1, b_2, b_3 \rangle$, then the cross product of $\overrightarrow{a}$ and $\overrightarrow{b}$ is the ...
2
votes
2answers
30 views

Finding $P$ knowing $\overrightarrow{PQ}×\overrightarrow{b}$, $\overrightarrow{PQ}⋅\overrightarrow{c}$, $\overrightarrow{b}$, and $\overrightarrow{c}$

Let $Q$ be the point $(1,2,3)$, let $\overrightarrow{b} = \langle -1, 0, 1\rangle$, and let $\overrightarrow{c} = \langle 2, 1, 5\rangle$. It is known that $\overrightarrow{PQ} \times ...
3
votes
2answers
47 views

Sum of three cross products is zero.

Let $u,v,w\in \mathbb R^3$. Prove $u \times( v \times w)+v \times( w \times u)+w \times( u \times v) =0$ I guess things would work out if I just expanded as a ton of products. Is there a better way?
0
votes
0answers
37 views

Properties of vectors in $\mathbb{R}^3$ related by cross-products

If $u, v$ and $w$ are vectors in $\mathbb{R}^3$ such that $u \times w=v \times w$, which of the following will be true: a) $\pi_w u= \pi_w v$ b) $\pi_u w= \pi_v w$, where $\pi_x$ is the projection ...
1
vote
2answers
58 views

Use the cross product to find a parallel vector

I'm confused by this exercise here : Using the cross product, for which value(s) of t the vectors w(1,t,-2) and r(-3,1,6) will be parallel I know that if I use the cross product of two vectors, I ...
0
votes
1answer
14 views

A plane and the matter of vector crossing order

I have three 3D points $A$, $B$ and $C$ which are defining a plane. If I want to get the equation of the plane, firstly I need its normal vector. Is it matter if I do it with $AB \times AC$ or $AC ...
0
votes
1answer
38 views

Why is the cross product a x b dependent on size of vector a?

If I consider a x b = c, as a system where the vector b is rotating about an axis defined by vector a, and vector c shows the linear direction which vector b moves as it is rotating. The faster the ...