For questions on cross products.

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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 ...
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1answer
106 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 ...
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1answer
80 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?
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43 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 ...
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1answer
27 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 ...
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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 $$ ...
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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 ...
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2answers
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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 ...
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1answer
111 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 ...
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1answer
48 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 ...
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1answer
41 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 ...
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37 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 ...
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37 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 ...
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1answer
89 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 ...
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1answer
36 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 ...
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2answers
29 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 ...
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2answers
76 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 ...
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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 ...
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1answer
46 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 ...
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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 ...
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4answers
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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 ...
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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?
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2answers
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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 ...
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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 ...
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2answers
49 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?
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38 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 ...
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2answers
78 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 ...
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1answer
15 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 ...
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1answer
39 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 ...
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1answer
32 views

Can you relate the cross product of two vectors to rotational motion.

Say you have a cross product a x b = c. Can you intepret this as the vector b spinning about the vector a. If the angle between them is close to zero then vector b is close to zero and is spinning ...
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What is the intuition behind the cross product? [duplicate]

I've researched lots of places and still cannot wrap my head around the cross product. The closest thing I have to an understanding of the cross product is that its a measure of how orthogonal two ...
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1answer
714 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 ...
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1answer
27 views

How to plot a contour in the plane defined by a vector cross-product

I have some contours of a structure in 3D that I would like to plot in 2D, but how do I get from 3D to 2D? In other words, I would like to plot the contour in the plane defined by the cross-product of ...
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28 views

cross product of material derivative

I am looking to evaluate $\vec{n} \times \dfrac{D\vec{u}}{Dt}$ where $\dfrac{D}{Dt}$ is the material derivative. Can I bring the cross product into the derivative and rewrite the expression as ...
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4answers
76 views

Equation perpendicular to 2 non-parallel planes

Good day sirs! Can you help me with this questions? Find the general equation of the plane: (1) Through $(3,0,-1)$ and perpendicular to each of the planes $x-2y+z=0$ and $x+2y-3z-4=0$ (2) ...
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1answer
97 views

Intersection of Three Planes proof

I'm supposed to be making a study guide answer for this question, but I'm struggling with proof. Show that the three planes intersect at the point provided that Note that the ...
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2answers
3k views

Using the cross product to find the angle between two vectors in $\Bbb R^3$

Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle$$. Find the angle between $u$ and $v$, first by using the dot product and then using the cross product. ...
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1answer
40 views

Understanding step in deriving the formula of the curvature.

Last formula on third page of the document: Computation of $\vec{r'}(t)\times \vec{r''}(t)$ From the previous two formulas and using the properties of cross products we see that ...
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1answer
56 views

Dot product of two vectors obtained by cross product

How can I prove the following identity: $$(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}) = (\vec{c}\cdot\vec{a})(\vec{b}\cdot\vec{d})-(\vec{b}\cdot\vec{c})(\vec{a}\cdot\vec{d})$$
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3answers
532 views

Cross product spherical coordinates

I can't wrap my head around the result of the cross product of two vectors in spherical coordinates. Is it a vector or something that I can represent geometrically? For example, given two vectors in ...
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77 views

Scalar triple product of quaternion scalar parts

I'm reading this paper about quaternion and 3D rotation with unit quaternions, \begin{eqnarray*} && \dot{q} = (q, {\bf q}) \\ && \dot{r} = (r, {\bf r}) \\ && \dot{r'} = ...
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Are all 7-dimensional cross products isomorphic?

Let $\times$ be this 7-dimensional cross product and let $\hspace{.04 in}f$ be a bilinear map on $\mathbb{R}^7$ which satisfies the orthogonality and magnitude conditions. Does there necessarily ...
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3answers
330 views

Making sense of a cross product of three vectors

Because of the cross product of two vectors being another vector I can calculate $\vec a\times(\vec b\times\vec c)$ as well as $(\vec a\times\vec b)\times\vec c$. I know that the cross product is not ...
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2answers
75 views

Using Gram-Schmidt to compute the cross product of $3$ vectors in $\Bbb R^4$ [duplicate]

I want to ask about vector multiplication (cross product) in $4$-d. I heard that Gram-Schmidt process is involved but I am not sure how the process is involved. The multiplication involves $3$ ...
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3answers
47 views

cross product of vector and direction

We know that cross product gives a vector that is orthogonal to other two vectors. Let this vector denoted by $$|\vec{v} \times \vec{u}| = \vec{n}$$ Then $$\vec{n}\cdot \vec{u} = 0 $$ Everything okay ...
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1answer
64 views

Definition of the vector cross product

As far as I understand the cross product between two vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$ is defined as a vector $\mathbf{c}=\mathbf{a}\times\mathbf{b}$ that is orthogonal to the plane ...
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1answer
60 views

Is seven-dimensional cross product rotationally invariant?

For three-dimensional cross product, the following property holds true: \begin{equation} (R\mathbf x) \times (R \mathbf y)=R(\mathbf x \times \mathbf y) \end{equation} where $R\in SO(3)$. Is the ...
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1answer
29 views

Differential Equation with Cross Products [without separating into system of equations]

I need to solve the following equation: $$ \frac{d m}{d t}=-m\wedge b-\alpha m\wedge (m\wedge b), $$ where $b$ is constant However, I was instructed specifically not to separate the calculation into ...
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3answers
34 views

The magnitude of a triple product of two vectors

So I was going through a past exam for electrodynamics and a question for radiation came up and within it was the following magnitude of a triple product $ \lvert \hat{r} \times [\hat{r} \times ...
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2answers
33 views

Find Orthogonal Vector's Peak Point

I am given a 3-component vector $\vec v$. There are obviously an infinite number of orthogonal vectors to $\vec v$. I need to find the specific orthogonal vector, lets call it $\vec{x}$, in the plane ...