For questions on cross products.

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0
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1answer
32 views

Shortest distance proof

Show that the shortest distance from a point $P$ to the line through $P_0$ with direction vector $\overrightarrow{d}$ is $$\frac{|\overrightarrow{P_0P}\times ...
0
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2answers
40 views

Dot and Cross Product Proof: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$

How do you prove that: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ ? The textbook says as a hint to "first do it for $u=i,j$ and $k$; then write $u-xi+yj+zk$ but I am not sure what that ...
-1
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0answers
19 views

Shortest distance and Cross Product [on hold]

Show that the shortest distance from a point P to the line through Po with direction vector d is $$ ||P_oP \times d||/||d||$$. I need help writing the proof for this. So far I have: let $ ...
0
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1answer
12 views

Cross Product and Scalar Product proof

Hi i'm trying to prove the following equality where P,Q and R are any 3D vectors: PxQxR = (P.R)Q - (Q.R)P I find it easier by proving that the x coordinate of the left side is equal do the right ...
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2answers
92 views

Proof of identity: cross product of three vectors

A book I'm reading contains the following (paraphrased) \begin{equation} (a \times b) \times c = (a \cdot c)b - (b \cdot c)a \end{equation} This is supposed to follow from: \begin{equation} (a \times ...
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2answers
25 views

Manipulation of three vectors in $\Bbb R^3$

Please excuse the non-specific title, this is a rather long problem. So on our last exam in multivariable calculus, our professor gave us a very lengthy vector manipulation problem as a bonus. Seeing ...
1
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3answers
33 views

Intuitively, what makes two vector parallel

I have heard explanations such as it is when the cross product equals zero or that it is when one is a scalars multiple of the other but I have not seen an intuitive explanation. Is it when two ...
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0answers
28 views

Systems of equations of the form $\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a$

Is there any theory that deals (directly or not) with systems of equations of the form $$\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a,$$ where $a \in \mathbb{R}^3$ is known, $v_i, v_j \in ...
0
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1answer
44 views

Cross product for vector angular position?

The angular velocity of a particle $\omega = r \times v$ is a pseudovector because it is formed by the cross product of two vectors (position and linear velocity). Likewise the angular acceleration ...
3
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8answers
239 views

Why is the cross product of two vectors orthogonal?

What is the intuition behind the fact that the cross product of two vectors is orthogonal? Every video I've seen just says it is orthogonal but they do not explain why. Since I have terrible memory, I ...
0
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1answer
33 views

Cross product in uneven matrices

I don't need help with dot product, only the cross product section. Even a hint as to where to start would be great.
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0answers
28 views

Is this tensor identity true?

If We have two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ and a symmetric positive definite Matrix $\boldsymbol{M}$ I was wondering if the expression $((\boldsymbol{a}\times \boldsymbol{b}) \cdot ...
0
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1answer
33 views

Linear Algebra: Compute Area of Parallelogram

I have this one Linear Algebra question that is asking me to compute the area of a parallelogram defined by 4 vectors. Here is the question: Let $\vec{u}=\begin{bmatrix}a\\b\end{bmatrix}$ and ...
1
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2answers
45 views

vectors in $3$ dimensions such that $v+(v \times u)=u$

Let $i,j,k$ denote the usual three unit vectors in $\mathbb R^3.$ 1) Find all vectors $v \in \mathbb R^3$ such that $v+(v \times i)=j$. 2) Suppose vectors $v$ and $u$ belong to $\mathbb R^3$ and ...
1
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0answers
34 views

Integration of multivariate Gaussians with cross terms

I'm stuck with the following integral: $I=\int ... \int exp\Big(-\frac{1}{2} \sum \limits_{t=1}^{n} x_{t}^T{\Sigma_{x}}^{-1} x_{t}+\sum \limits_{t=1}^{n} x_{t}^T{\Sigma_{x}}^{-1} z_{t} -\frac{1}{2} ...
1
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1answer
34 views

Problem with understanding a cross product related problem

I have a question that follows like this: Let $U = (0,1,2)$ and $V = (1,-1,-2)$. Assume that $V \times W = (3,1,1)$ Is there enough information provided in order to determine $(U \times V) ...
0
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2answers
31 views

Cross Product of two vectors

I have these two vectors $$\vec{a}_1=\frac{a}{2}\hat{x}+\frac{\sqrt{3}}{2}a\hat{y}$$ $$\vec{a}_2=c\hat{z}$$ I know $\vec{a}_1\times\vec{a}_2$ is equal to: ...
0
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2answers
31 views

Question on Vectors

Given $a = [-5, 8, 1]$ and $b = [2, -7, -3]$, find a vector $c$ such that $a \cdot (b × c) = 0$ I don't know how to get it, I've been looking for examples, but I don't know..
0
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2answers
29 views

Angular velocity

The angular velocity $\omega$ of rotation of a rigid body has the direction of the rotaion axis and magintude equal to the rotation rate in rad per second. The orientation of $\omega$ is determined by ...
0
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1answer
31 views

Torque of a force

The torque $M$ of a force $\overrightarrow{F}$ as for the point $O$ is defined as the product of the magnitude of the force $\overrightarrow{F}$ and the perpendicular distance of the point $O$ and the ...
0
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1answer
30 views

Vector cross product properties?

I have a 3x3 symmetric matrix $$C=AB^T+BA^T,$$ where both $A$ and $B$ are 3x1 vectors. How may I prove $$C(A\times B)=0?$$ I believe the key is the properties of vector cross product.
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1answer
43 views

Curiosity with the Cartesian Notation of the Vector Cross Product

In my opinion Hibbeler's book on statics (Engineering Mechanics Statics, 12th ed) is one of the most approachable on the subject. On pg.123 he defines the Vector Cross Product in its Cartesian ...
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5answers
59 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 ...
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6answers
66 views

Unit vectors that are orthogonal to vectors

I have to find all the unit vectors that are orthogonal to the vectors $\overrightarrow{a}=(2, -4, 3), \overrightarrow{b}=(-4, 8, -6)$ . I calculated that the cross product $\overrightarrow{a} ...
0
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1answer
37 views

Dot product of two cross products in $\Bbb R^3$ with general metric

I would like to find the generalized formula of the identity $$(A\times B).(C\times D)=(A\cdot C)(B\cdot D)-(A\cdot D)(B\cdot C)$$ which holds in an Euclidian metric, within a general metric $g$ on ...
0
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0answers
18 views

I have problem in understanding the relation?

If we have a relation like this: $$\frac { \partial }{ \partial x_\beta } (\varepsilon_{ij\alpha} x_j T_{\alpha \beta}) =\left[ \nabla \cdot(\vec x \times \overset {\leftrightarrow}{T} ) \right]_i$$ ...
1
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2answers
25 views

Algorith/ Equation to get the ith element in N x N

I am having a difficulty figuring out the equation to get the ith element in $\mathbb{N}\times \mathbb{N}$ ( crossing the set of natural numbers).We have $\mathbb{N}\times \mathbb{N} = \{ ...
1
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3answers
132 views

How can I prove that two vectors in $ℝ^3$ are linearly independent iff their cross product is nonzero?

Here's my attempt: Let $𝒙 = (x_1, x_2, x_3)$ and $𝒚 = (y_1, y_2, y_3)$ The cross product of $𝒙, 𝒚$ is $𝒙⨯𝒚=(x_2y_3-x_3y_2, x_3y_1 - x_1y_3, x_1y_2 - x_2y_1)$ And linear independence of $𝒙, ...
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1answer
56 views

Generalising the cross product to infinite dimensions, does $v \times v = 0$ hold also in infinite dimensional spaces

Consider I have a vector space $V$ with inner product and a bilinear map $b : V \times V \to V$ i) such that if $z = b(u,v)$ for two $u,v \in V$, then $$ z \perp u \quad \mbox{ and } \quad z \perp ...
1
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1answer
28 views

Points defining plane - starting step? [closed]

If the points $P, Q, R$, not all lying on the same straight line, have position vectors $a, b, c$ respectively, show that $(a \times b) + (b \times c) + (c \times a)$ is a vector perpendicular to the ...
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0answers
34 views

Cross Product for Biot-Savart Derivation of Current Loop

Biot-Savart's law can be used to determine the magnetic field produced by a figure at a point. Introductory physics texts integrate $dB$ to obtain $B$ where $dB$ = $\frac{I\mu_{0}}{4\pi r^2} * dl ...
2
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2answers
39 views

Ratio of area formed by transformed and original sides of a parallelogram

I am interested in finding the ratio of area formed by transformed and original sides of a parallelogram, given by: $$\frac{\|Ma\times Mb\| }{\| a\times b \|}$$ $M$ is a $3 \times 3$ matrix and $ a, ...
2
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1answer
34 views

Vector Cross Product and Expression for perpendicular distance between any two Vectors

If $B \ne C$, prove that the perpendicular distance from $A$ to the line through $B$ and $C$ is $$\dfrac {|| (A-B)\times(C-B)||}{||B-C||} $$ where $\times$ means the vector cross product. Attempt: ...
1
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1answer
20 views

Visualizing cross product of points in 3-Space

If $p_0, p_1, p_2$ are three distinct points in space, then what does the cross product $$n = (p_0 - p_1) \times (p_0 - p_2)$$ mean geometrically? I'm having a little trouble visualizing this in ...
0
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2answers
25 views

How to find an orthogonal vector C in $C^3$ relative to two other (given) vectors?

$A = [2,1,-i]$ $B = [i, -1, 2i]$ I need to find a C that is orthogonal to A and B. I've tried taking AxB, but this does not work. I get the vector C = (i, 1-4i, -2-i). The problem is that ...
9
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2answers
133 views

Is there a deeper meaning behind the “determinant” formula for the cross product?

We all know that for all vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R^3}$, if $(a_x,a_y,a_z)^\top$ is the component form of $\mathbf{a}$ and similarly $(b_x, b_y, b_z)^\top$ is the component form of ...
2
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2answers
63 views

Exterior Product vs Cross Product

I was confused about the relationship between a set of basis vectors in 3D, $ \left\{\hat e_1, \hat e_2, \hat e_3 \right\} $ and their exterior products. In my head, it makes sense that the identity ...
0
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4answers
26 views

Why we can conclude immediately that $x \in B$, if $(x, y) \in A \times B = B \times A$

The following statements are part of a proof involving cartesian products, specifically involving this theorem: $A \times B = B \times A \iff$ either $A = \emptyset$, $B= \emptyset$, or $A = B$ ...
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2answers
81 views

Relation between volume form and cross product

Euclidean three-dimensional space (it's simpler). Defining $\eta={e^*}^1 \wedge {e^*}^2 \wedge {e^*}^3$, with $\{{e^*}^1,{e^*}^2,{e^*}^3\}$ dual of the orthonormal basis, and indicating the classic ...
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0answers
58 views

Using the exchange lemma provides weird result

I know that if we have a square $n \times n$ matrix $A$ in F and two vectors $v,u$ in $\mathbb F^n$, following the exchange lemma we can show that: $$\langle Au,v\rangle = \langle u, \ A^*v\rangle ...
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1answer
135 views

Can the cross product be a matrix?

On the Wikipedia article on the cross product is says that a vector $a$ which is itself a cross product (that is $a=c\times d$), can be represented in the expression $a \times b$ for some other vector ...
1
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1answer
97 views

Integrating a cross product.

I am given that $\mathbf{k} . m \mathbf{q} \wedge \mathbf{\ddot{q}} = 0$ and my book says that integrating this wrt time gives $\mathbf{k}.m \mathbf{q} \wedge \mathbf{\dot{q}} = $constant. I don't ...
0
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2answers
57 views

How do you keep track of what vectors nabla ($\nabla$) should be working in on?

Take the following example: $$\vec\nabla\times(\vec A \times \vec B)$$ I assumed that this worked out to: $$\vec A(\vec\nabla.\vec B) - \vec B(\vec\nabla.\vec A)$$ Where, in both terms, Nabla ...
0
votes
1answer
17 views

Cross product problem

someone could show me the error in the cross products? For $U=x\hat{i}+y\hat{j}+z\hat{k}$, $V=x'\hat{i}+y'\hat{j}+z'\hat{k}$ and $((.))$=modulus, we have $$U \times V=((U))((V))sin(U,V).n = ...
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2answers
43 views

Bilinear form and cross product in hyperbolic geometry

I'm reading Patrick J. Ryan's Euclidean and non-Euclidean geometry, page 152. There is a bilinear form defined by $b\left( {x,y} \right) = {x_1}{y_1} + {x_2}{y_2} - {x_3}{y_3}$ on ${\mathbb{R}^3}$ and ...
0
votes
1answer
46 views

Cross product of two vectors, given magnitudes and angle

Problem Two vectors $\mathopen|{\overrightarrow{a}|=5.39} \ and \ \mathopen|{\overrightarrow{b}|=4.65} $ intersect and make a 120° angle. Find $\mathopen|{\overrightarrow{a}}\times ...
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0answers
9 views

Find coordinates of point that satisfy given conditions

I have A(1,2,3) , B(-1,0,1), C(1,-1,1) which are points in $\mathbb{R}^3$. I'm trying to find another point H such that AH${\parallel}$AC and BH${\perp}$AC. I set H = ($h_1$, $h_2$, $h_3$), and took ...
1
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0answers
23 views

Simple indefinite integral of a vector function

I am having trouble with this simple integration. I am not sure of the process or steps to follow to solve this type of problem: If $\mathbf{V}(t)$ is a vector function of $t$, find the indefinite ...
0
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0answers
19 views

Axes of rotation, recursive tree branching and GLrotate (computer graphics)

The question is to solve a computer graphics problem, but is essentially a vector math problem so I think it belongs here. My problem is this: a recursive tree is being generated for n iterations ...
0
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2answers
31 views

Vector equation with cross product and unit vector

Does anybody know how to solve the equation $\mathbf{a} + \mathbf{b} \times \hat{\mathbf{v}} = c \hat{\mathbf{v}},$ where $\mathbf{a}$ and $\mathbf{b}$ are given real vectors, for the unit vector ...