1
vote
1answer
57 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 ...
0
votes
2answers
51 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 \\ ...
1
vote
2answers
63 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 ...
2
votes
1answer
27 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 ...
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) ...
1
vote
1answer
39 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 ...
3
votes
1answer
177 views

Name of an identity for traceless matrices in $\mathbb{R}^3$?

While working on a more compact presentation of a derivation in the context of incompressible fluid flow we tried to simplify things by introducing intermediate steps instead of writing out lengthy ...
3
votes
1answer
34 views

Cross Product in Levi-Civita Notation - The elementary basis vector's missing?

http://www.unl.edu.ar/ceneha/uploads/Cartesian_tensors_Index_notation_&_summation_convention.pdf avers: $1.$ $(a×b).(c×d) = \epsilon_{i jk}a_jb_k \quad e_{ilm}c_ld_m$ $2. \nabla × ...
0
votes
2answers
50 views

Adding two vectors such that the resulting vector is perpendicular to a third vector

Let $$a = (-3, 3, 1)$$ $$b = (1, 4, -4)$$ $$c = (2, 1, -3)$$ For which values of $t \in \Re$ is $b + tc$ perpendicular to a? For a vector to be perpendicular to $a$, the dot product of that ...
0
votes
0answers
102 views

What is does the transformation $[\mathbf{a}]_{\times}$ do?

https://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication I'm curious about the matrix $[\mathbf{a}]_{\times} \stackrel{\rm def}{=} ...
0
votes
2answers
48 views

Prove that vectors x,y are linearly dependent exactly when …

Prove that vectors $\vec{x},\vec{y}$ (belonging to $\mathbb{R}^3$) are linearly dependent only if the following is true $$ \begin{vmatrix} x_1&y_1 \\ x_2&y_2 \end{vmatrix} ...
4
votes
0answers
132 views

Cross Product - Moments :: Dynamics

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the ...
0
votes
2answers
283 views

Finding the volume of a pyramid (the vector way)

The problem I have 4 points $ P \; (-1,2,0) \\ Q \; (2,1,3) \\ R \; (1,0,1) \\ S \; (3,-2,3) $ and I want to find the volume of a pyramid. What I'm most concerned here is the appropriate strategy ...
0
votes
2answers
165 views

Proof in a Scalar Triple Product [closed]

For any three vectors $\vec a,\ \vec b,\ \vec c$, show that : $$[\vec a\times\vec b,\ \vec b\times\vec c,\ \vec c\times\vec a]=[\vec a,\ \vec b,\ \vec c]^2$$ where $[\vec a,\vec b,\vec c]=\vec ...
0
votes
1answer
33 views

Given 3 cross products of 3 vectors, how do you solve an expression of this format?

If you're given: $$a \times b = (2, -4, 2),\quad a \times c = (7, 13, -11),\quad b \times c = (1, 7, 1)$$ what properties of cross products or formulas can you use to solve $(2b - c) \times (3a + ...
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, ...
3
votes
1answer
99 views

Show that $e^{\theta(s\times)} = I + \sin\theta(s\times) + (1 − \cos \theta)(s\times)^2$

$$e^{\theta(s\times)} = I + \sin\theta(s\times) + (1 − \cos \theta)(s\times)^2$$ I have to prove the above formula and am not sure where to start, may someone please help me! The full question is ...
0
votes
3answers
53 views

Help proving that $\vec{a}\times(\vec{a}\times(\vec{a}\times\vec{b}))\cdot\vec{c} = -\|\vec{a}\|^2\vec{a}\cdot\vec{b}\times\vec{c}$

This is problem 13 from Chapter 13, Section 14 of Apostol's Calculus Vol 1. I need to prove or disprove the formula $\vec{a}\times(\vec{a}\times(\vec{a}\times\vec{b}))\cdot\vec{c} = ...
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 ...
7
votes
1answer
242 views

Does this cross-product norm inequality hold?

Let $\times$ denote the cross-product. $\;$ Is it the case that For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in ...
8
votes
5answers
3k views

Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
2
votes
1answer
155 views

Multiplying vectors (answered own question)

I recently realised that asking a question and answering our own question is allowed here, so here is a question I've seen commonly on many sites: "How does one multiply two vectors?" This is very ...
1
vote
1answer
95 views

Find closest vector to A which is perpendicular to B

To start, I would like to apologize if the answer to my question was easily googled, I am quite new to this and googling "Find closest vector to A which is perpendicular to B" gave me no results. My ...
0
votes
1answer
91 views

Cross product as factor in dot product [duplicate]

Given there are two vectors $w,v$ with $||w||=4$ , $||v||=1$ and $\phi=\frac{2\pi}{3}$ How do you transform the following expression into a form in which it can be computed with the given ...
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?
3
votes
2answers
173 views

Help over the proof of triple vector product identity

For all vectors $\bf{x}$, $\bf{y}$ and $\bf{z}$, $$\bf{x}\times(\bf{y}\times\bf{z})=(\bf{x}\cdot\bf{z})\bf{y}-(\bf{x}\cdot\bf{y})\bf{z}$$ The proof goes as follows: We may suppose that $\bf{y}$ ...
-3
votes
2answers
148 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
3answers
6k views

Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.

I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$.
1
vote
1answer
154 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
182 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 ...
0
votes
1answer
214 views

Cross product as result of projections

The cross product between two vectors in $\Bbb{R}^3$ (call them a and b) is denoted a $\times$ b and the result is a vector in $\Bbb{R}^3$ orthogonal to the first two. There are a variety of ways of ...
6
votes
2answers
142 views

Explanation of a cross product result

In my book the result $$(u\times v)\cdot(x\times y)=\begin{vmatrix} u\cdot x & v\cdot x \\u \cdot y & v \cdot y\end{vmatrix},$$ where u, v, x and y are arbitrary vectors, is stated (here ...
5
votes
3answers
96 views

How to divide by $(a_1,a_2,a_3)$

I have been searching for an explanation in Howard's Linear Algebra and couldn't find an identical example to the one below. The example tells me that vectors $\boldsymbol{a}_1$, $\boldsymbol{a}_2$ ...
1
vote
1answer
1k views

Cross product in complex vector spaces

When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors? I suppose that one possible ...
4
votes
2answers
133 views

Why is $\det(\vec{A},\vec{B}) = |\vec{A} \times \vec{B}|$?

In the multivariable calculus class the teacher showed us the formula of the cross product $$ \vec{A} \times \vec{B} =\begin{vmatrix}\hat{\imath}& \hat{\jmath}& \hat{k} \\ a_1 & a_2 ...
16
votes
4answers
3k views

Origin of the dot and cross product?

Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...
2
votes
1answer
295 views

Minimize sum of the norm of cross products

Here I have an interesting problem on linear algebra. It looks very simple, but not so easy to solve for me. Let $r_i, i=1,…,n$ be unit vectors in $\mathbb{R}^n$, find a unit vector $x$ to minimize ...
11
votes
4answers
4k views

Wedge product and cross product - any difference?

I'm taking a course in differential geometry, and have here been introduced to the wedge product of to vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) ...