For questions on cross products.

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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 ...
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17 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 ...
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0answers
7 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 ...
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2answers
26 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 ...
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1answer
20 views

In this situation, based on order of operations, would cross product happen first or dot product?

I got from wikipedia that the dot product is also referred to as the "scalar product" and that the cross product is also referred to as the "vector product". Can anyone confirm my inference on the ...
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'Simple' Vector analysis and I need a little help prooving an identity.

I have to prove that $$\vec{S}(\vec{r}) = -\frac{c^2}{\omega} \bigg( u(\vec{r})\vec{\nabla}\phi(\vec{r}) + \frac{i}{2}\vec{\nabla}u(\vec{r}) \bigg) ~~~~(1)$$ given that $$\vec{S}(\vec{r}) = ...
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47 views

Backward Euler method with a cross-product.

I want to solve the following differential equation with the backward Euler method ...
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Using cross product to determine which way to rotate

I'm trying to make a car go up a slope. So I have this snippet of code to turn the car when it collides with the ramp. I'm currently using cross product of forward vector of car with the direction ...
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1answer
32 views

using a symmetric matrix to simplify expression involving antisymmetric matrices

I am trying to simplify the following expression --$$(\vec{A} \times \vec{B})\cdot\underline{\underline{S}}\cdot(\vec{C} \times \vec{D}) = (A_jB_k\epsilon_{jki})S_{im}(\epsilon_{mnp}C_nD_p)$$ $S$ is a ...
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2answers
35 views

Is there a relationship between the cross product and quaternion multiplication?

I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus). Motivation for the question: I've been introduced to ...
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1answer
63 views

Cross Product in $\mathbb{R}^n$

I know that the cross product only exists in $\mathbb{R}^3$ and $\mathbb{R}^7$ but I am wondering what the actual definition of cross product is. That is, a cross product would be a function ...
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1answer
14 views

Find a plane defined by a point, a ray, and a vector starting from the point and parallel to another plane

I am trying to figure this out for implementation into a Graphics manipulator I've been trying to work out. I need to find a plane (a normal vector to the plane will suffice) and I know some of its ...
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2answers
68 views

Question about cross product and tensor notation

I am a bit rusty on tensor algebra and calculus and may use some wrong terminology, but I know that the cross-product can be expressed in tensor notation with the aid of the ...
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2answers
78 views

The formula for the magnitude of cross product, $\| u\times v \| = \| u \| \|v \| \sin \theta $ [closed]

Can someone show me a proof of the magnitude (length) of the cross product: $$\|u \times v \| = \| u \| \|v \| \sin \theta $$
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1answer
36 views

Cross product simplification

If you have two vectors, $A$ and $B$, then we can write the cross product as $A \times B$. Simplify the following expressions: $A \times (A \times (A \times B))$ $A \times (A \times (A \times (A ...
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0answers
22 views

Calculate a percentage based on intervals

I'm trying to calculate the relevance of an object based on it's location and a user's desired max distance. Also, I've relaxed the DB constraints, and retrieved from DB, objects that are twice as far ...
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1answer
28 views

Solving for first term in vector product

I'm trying to solve a system of equations for a physics application I've been working on, and I'm down to one thing left that's stumping me. Essentially, I need to solve $$A \times B = X$$ where $A, ...
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2answers
71 views

Proving a cross product satisfies the vector equation

Let vectors $u,v,w \in R^3$ Prove that $u \times (v \times w)$ must be a vector that satisfies the vector equation $x=sv+tw$ where $s,t \in R$ I have no idea where to go with this one, any tips?
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22 views

Proof dealing with orthogonality and cross product

If $v$ is orthogonal to vectors $x$ and $y$, then $v$ is a scalar multiple of $x \times y$. So far I have that: $v\cdot x=v_1x_1+v_2x_2+v_3x_3=0$ and $v\cdot y=v_1y_1+v_2y_2+v_3y_3=0$ $x \times y ...
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0answers
30 views

The identify with cross product : how to understand?

The problem come from the notations in Hansbo's paper (page 197) and his another paper. Given a domain $\Omega$, the function $u\in H_0^1(\Omega)$. $\Omega$ is divided into 2 subdomains ...
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1answer
25 views

Show that Two Vectors Making Supplementary Angles?

I just need a start. I am not looking for whole prove, but it'd be more appreciated if I get one. Q. Use Theorem u . v = |u| |v| cos a and the trigonometric identity, cos (180-a) = -cos a, to ...
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0answers
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Proving formula for length of crossed product

Suppose we are working in $\mathbb{R}^3$ and have the defined the usual scalar product and proven Cauchy-Schwartz. Then we can define the angle, $\theta$, between non-zero vectors by requiring $a\cdot ...
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1answer
27 views

$\nabla \times \underline{v}$ - Results in a vector perpendicular to these two vectors?

Say $v = -y\hat{i} + x\hat{j}$ If we take the cross product of $\underline{v}$ with $\nabla$ we get $\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{d}{dx} & \frac{d}{dy} ...
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1answer
36 views

If $u$ and $v$ are vectors in $3$-space, then $u\cdot v$ is a scalar

My understanding is that B is definitely true because of the below picture but I cannot understand A. Please would someone point me to the right direction! Thanks!
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3answers
45 views

Where does the right hand rule appear in the “tensor” definition of the cross product?

When doing the cross product of two vectors according to the usual geometric definition of $\mathbf{A}\times\mathbf{B}$ being perpendicular to both $\mathbf{A}$ and $\mathbf{B}$, it's pretty clear ...
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1answer
50 views

how $3i \times 3i = 9i \times i$? (i is the unit vector and $\times$ is cross product)

$i$ is the unit vector; didn't know how to write it. I'm reading a text and somewhere it uses something like $ai \times bi = (ab)i \times i$ (implicitly). I can see why this is true geometrically, ...
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1answer
54 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
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1answer
43 views

Vector cross product identity for $(a\times b)\cdot(c \times d)$

Prove that $(a\times b)\cdot(c \times d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)$ I would appreciate some hints on how to solve this.I assume there is a method which does involve equating LHS ...
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5answers
510 views

Sphere equation given 4 points

Find the equation of the Sphere give the 4 points (3,2,1), (1,-2,-3), (2,1,3) and (-1,1,2). The *failed* solution I tried is kinda straigh forward: We need to find the center of the sphere. Having ...
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1answer
46 views

Is there any associative algebra that has all the algebraic properties of cross product?

Cross product is not associative, but is there any non-trivial associative algebra that has all the algebraic properties of cross product except the condition that would violate associativity and ...
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1answer
74 views

Is the matrix form of the cross product related to bilinear forms.

The cross product of two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$ can be represented as a matrix product as follows, if $\mathbf{x} = (x_1, x_2, x_3)^{\top}$ then $\mathbf{x} \times ...
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1answer
23 views

Cross-product is a left singular vector?

Assume A is a 3x2 matrix with rank(A)=2. u1 and u2 are already left singular vectors... How would I go about proving that the cross-product of the two is also a left singular vector? Hints would be ...
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0answers
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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 ...
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Cross product query.

In cross product, we do it like: a vector x b vector = a*b*sin(theta). From where does this sin(theta) came from? Can someone please derive the cross product and explain it.
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covariance of cross product of two vectors

I have two independent vectors in 3D and know the covariance matrix of each. What will be the covariance of cross products of above vectors. In particular what will be the covariance of cross ...
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1answer
188 views

Product rule for gradient of cross product

The book I am reading gives a list of product rules, among them the following: $$\nabla \cdot (v\times w) =(\nabla \cdot v) w-v\nabla \cdot w.$$ However, the left-hand side is a number whereas the ...
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1answer
63 views

Programming constraints in video game. How are these two equations equal?

I'm currently working on programming a game that uses a physics engine (NAPE). Inside of that engine there are constraints that you can program. In order to program those you need a somewhat ...
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1answer
41 views

If $u$ and $v$ are vectors in $R^3$, simplify the expression $(u+v) \times (u-v)$ as much as possible.

Here is my thinking process for answering this question: Cross product is neither commutative nor associative. Hence I cannot do any algebraic operations on this expression. However I know that cross ...
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2answers
112 views

How to prove this Gram determinant

Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$ \det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ...
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3answers
43 views

Calculus - Components of a unit vector

Determine the components of a unit vector perpendicular to (0, 3, -5) and (2, 3, 1). I think I should be using either cross or dot product, but am unsure on what to do from there.
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2answers
341 views

Given two unit vectors, find a vector perpendicular with additional constraint

Given two unit length vectors find a perpendicular vector of unit length. I want to know if there's a way to do this without using a square root operation (avoid a normalization operation). Since the ...
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1answer
367 views

cross product in cylindrical coordinates

Hi i know this is a really really simple question but it has me confused. I want to calculate the cross product of two vectors $$ \vec a \times \vec r. $$ The vectors are given by $$ \vec a= a\hat ...
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1answer
141 views

Clarify Right hand Rule

I was just wondering whether in the right hand rule are all 3 vectors perpendicular to one another or is it simply one way, i.e. $A \times B=C$, would it be right to also say $C \times A=B$ and $C ...
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2answers
47 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 ...
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1answer
36 views

Cross Product in 3D

Hi! I am currently working on some calc2 online homework problems concerning the cross product. I understand how the cross product works, but I am not sure how to apply it to this question. I know ...
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1answer
40 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
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1answer
49 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) ...
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0answers
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Relational Algebra Cross Join Definition. Need Clarification

I'm currently reading Lee Wilkinson's Grammar of Graphics, and I'm having trouble understanding his mathematical definition of the cross join. I understand the input and output, but the relational ...
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1answer
145 views

How do I solve $F = \nabla\times G$ for $G$?

Given the vector valued function $F(x,y,z) = (xz,-yz,y)$ find $G$ such that $F = \nabla\times G$ I let $G(x,y,z) = (G_1,G_2,G_3)$ and expanded $\nabla \times G$ then equated the components to $F$ but ...
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1answer
50 views

Why is the cross product contained in orthogonal complement?

Let $(V,\langle,\rangle)$ be the $\mathbb R^3$ with the standard bilinear-form and let $W \subset V$ be a two dimensional spanning set given by $v = (x_1,x_2,x_3)$ and $w = (y_1,y_2,y_3)$ and the ...