Dot Definition and 552 Threads

Homework Statement Show that the dot product in two-dimensional space is linear: <u|(|v> + |w>) = <u|v> + <u|w> The Attempt at a Solution I feel like I'm missing some grasp of the concept here ... I would think to just distribute the <u| and be done in that one step, but I'm being...

I have been asked to solve the equation for x - All letters bar λ are vectors. λx + (a cross x) = b I have worked it down as far as x.(λb + (b cross a)) = |b|^2 by taking the dot product with b of both sides. But is there any way I can now solve this equation for x?

Homework Statement That is prove that |a•c|≤|a||c| for any vector a=<a1,a2,a3> & c=<c1,c2,c3> Homework Equations The Attempt at a Solution I really don't have much of an attempt at the solution. I am not sure where to start. I can kind of justify it in my mind by saying the...

Ok, I'm going to be taking calc III next week, so I wanted to get a head-start by doing the MIT multivariable calculus opencourseware. While most of the material was easy, these proofs are really killing me. Here are two examples: Ex.1: Using vectors and dot product show the diagonals of a...

I'm so confused about finding an angle, theta in this illustration. With having three coordinate information, how can I calculate the theta using dot product? I would easily find the angle by using trigonometric formula if I ignore the z-axis. But I want to solve this problem with...

I was thinking, if exist a product (cross) between vectors defined as: \vec{a}\times\vec{b}=a\;b\;sin(\theta)\;\hat{c} and a product (dot) such that: \vec{a}\cdot\vec{b}=a\;b\;cos(\theta) Why not define more 2 products that result: \\a\;b\;sin(\theta) \\a\;b\;cos(\theta)\;\hat{d} So, for...

Homework Statement Evaluate the line integral of F dot dr where f(x,y)=<3x^2,2x+y> and C is a straight line segment from (1,2) to (5,4) Homework Equations Unfortunately I was out with family obligations when we covered line integrals and surface integrals so am stuck with the textbook...

A simple question: what is the difference between inner product and dot product?

Hello, I have a quick question about integrals of dot products. We are learning about magnetic flux as the integral of b dot da. However, what circumstances must be present where we can simplify this integral into (b*a) and ignore the integral?

How does one change the dot product such that there is no dot product in between, just plain multiplication? For example, in the following: eb.\partialcea=-\Gammaa bc How do I get just an expression for \partialcea?

Homework Statement Prove that if u and v are nonzero vectors, and theta is the angle between them then u dot product v = ||u|| ||v|| cos (theta). Consider the triangle with sides u ,v , and u-v. The Law of Cosines implies that ||u-v||^2 = ||u||^2 + ||v||^2 - 2||u|| ||v|| cos(theta). On the...

can you show me derive cross product from dot product?

Hi, I am trying find the simplified expression of this: ∇(E \cdot E) Where E is the electric field that can written as E_{0}(exp(i(kx-ωt)) I know that since the two vectors are the same => E \cdot E = ||E||^{2} Do I take the gradient of the magnitude then? It just doesn't feel...

Homework Statement if v x w = <5,5,-2> (v cross w) and v * w = 6 (v dot w) then what is the tan(θ) between the two vectors v and w? The Attempt at a Solution well I was thinking v x w = |v||w|sinθ as well as v dot w (v*w) = |v|w|cosθ divide one equation by the other...

Homework Statement A constant force of 1i - 5j -8k moves (1,-4,2) (-3,2,-1), what is the work done on the particle? Homework Equations Avector*Bvector=ABsinθ ?? I think? The Attempt at a Solution I really am quite lost... but I found the coordinates for the position vector...

The problem is: Let A be a real m x n matrix and let x be in R^n and y be in R^m (n and m dimensional real vector spaces, respectively). Show that the dot product of Ax with y equals the dot product of x with A^Ty (A^T is the transpose of A). The way I went about starting this problem is to...

$$ \frac{(\dot{\mathbf{r}}\times\ddot{\mathbf{r}}) \times\dot{\mathbf{r}}}{\lvert\dot{\mathbf{r}} \rvert\lvert\dot{\mathbf{r}}\times\ddot{\mathbf{r}}\rvert} $$ How do I take that dot product of the expression of above with itself?

momentum, position vector dot (scalar) product "action" Hello, I was playing with single mass point classical mechanics, when I realized that the dot product of the position vector and momentum vector, p.r , has action dimension. Furthermore, its time derivative, d/dt(p.r) = F.r + p.v, has...

Homework Statement A=(x,3,1) ,B=(x,-x,2) Determine the value of x if the vector perpendicular to A and B is given by C=(10,-4,-4) Homework Equations The Attempt at a Solution Find A cross B , let A cross B be D . Then D cross C = zero (since they are perpendicular to both A and...

Homework Statement Does the Cauchy Schwarz inequality hold if we define the dot product of two vectors A,B \in V_n by \sum_{k=1}^n |a_ib_i| ? If so, prove it. Homework Equations The Cauchy-Schwarz inequality: (A\cdot B)^2 \leq (A\cdot A)(B\cdot B) . Equality holds iff one of the vectors...

Homework Statement Let A be a vector perpendicular to every vector X. Show that A = O Edit: it is O not 0. (OH not zero) haHomework Equations So, we know if A and X are perpendicular then A(dot)X = 0 I see no reason why A would have to be equal to 0. Could X (not equal) 0? Could it be...

Understanding the use of Pythagorean theorem for the length and square of a vector and how this is the dot product of a vector with itself is no problem. I'm trying to look inside the meaning of the dot product of two different vectors and understand it. I can also accept (just following the...

Is there a definition of quantum dot that everybody agrees? I searched intensively both from google and from web of science, but I couldn't find a formal definition. e.g (wiki says) A quantum dot is a semiconductor whose excitons are confined in all three spatial dimensions. confined but...

Problem: In Kleppner's book, Introduction to Mechanics, he states "By writing \vec{A} and \vec{B} as the sums of vectors along each of the coordinate axes, you can verify that \vec{A} \cdot \vec{B} = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}." He suggests summing vectors, but since the sum of two...

The centrifugal force is $$ \Omega\times r\times \Omega $$ I paper I am reading then writes it as ##\frac{1}{2}(r\Omega)^2 - \frac{1}{2}\Omega^2r^2## How was this obtained? Using the fact that ##a\times b\times c = (ac)b - (ab)c##, I don't get what they are getting so there is...

This seems like a very basic question that I should know the answer to, but in my image processing class, my teacher explained that a basis set of images(matrices) are orthonormal. He said that the DOT product between two basis images (in this case two 2x2 matrices) is 0. so, for example...

Hi all, I'm working on a math problem with a known answer - though I can't reproduce the maths. The problem is this: there is a random 3d vector of unit length with a uniform probability, \vec{v}, and a secondary unit vector \vec{u}. It is stated that: f = \int_{S^2}{| \vec{v} \cdot...

The definition of the dot product is given by A = <a1,b1> B = <a2,b2> A dot B = a1a2 + b1b2 Is this definition valid for orthogonal coordinates only?

R <x,y,z> A<a1,a2,a3> B<b1,b2,b3> Show that (r-a).(r-b)=0 represents a sphere find its center and radius So i see that if 2 vectors are orthogonal you can create a sphere and find the radius and center but can somone better explain this problem

find 2 unit vectors that make an angle of pi/3 with <3,4> <3,4>dot<a,b>=5/2=3a+4b b=5/8-3/4 a |<a,b>|=1 such that a^2+25/64+15/16a+9/16 a^2=1 25/16 a^2+15/16/ a=39/64 100a^2+60a=39 a^2+3/5a=39/100 (a+3/10)^2=48/100 a=(4sqrt(3)+-3)/10 so b=5/8-(12sqrt(3)+9)/40...

I need to determine what type of covalent bond SiC2 (silicon dicarbide) and so creating the lewis dot diagram is helpful. I know I have 12 electrons available but I can't seem to draw it :/

Find 2 unit vectors that make a 60 degree angle with <3,4> Vector <a,b> Taking b=1 cos60=1/2 3a+4=(5/2 )sqrt(a^2+1) 36a^2+96a+64=25a^2+25 11a^2+96a=-39 (sqrt(11)a+48)^2= 2265 +- a=(sqrt(2265)-48)/sqrt(11)

I'm reading up on dot products and keep seeing two different examples. One states that u\cdotv = u_{i}\cdotv_{i} + u_{j}\cdotv_{j} Another: u\cdotv = |u|\cdot|v|cosθ I'm not understanding when to use the first or second method?

I was recently going through the proof of Compton scattering and I saw that they took a square value and wrote it as p^2=p(dot)p= etc... Is this true or all squared values?

Homework Statement The Attempt at a Solution I am working a physics problem and want to make sure I'm not making a mistake in the math. Here is my math inquiry: Say you have (a*b)(c*d) where * indicates the dot product, and a,b,c, and d are all vectors. Can you say that (a*b)(c*d) =...

Homework Statement Let ##\gamma(t)## be a path describing a level curve of ##f : \mathbb{R}^2 \to \mathbb{R}##. Show, for all ##t##, that ##( \nabla f ) (\gamma(t))## is orthogonal to ##\gamma ' (t)##Homework Equations ##\gamma(t) = ((x(t), y(t))## ##\gamma ' (t) = F(\gamma(t))## ##F = \nabla f...

I was recently posed a riddle that went like the following: There are two people. Person A picks three numbers from 0-99. Person B guesses which three numbers that person A has picked. Then, person A gives the dot product of his picked numbers with person B's guessed numbers. The question is...

(maybe this should go in the math section?) Homework Statement I'm supposed to derive the k.p form of the Schroedinger equation by plugging in the Bloch wavefunction expansion. But my actual question is just about the math. Homework Equations So when I plugged Bloch into the SE, one of the...

By using both cross and dot products, the angle between 2 vectors can be found. But there is 1 question that I tried for countless times that the result of cross product and dot product are not the same. Here is the vectors that I am talking about A= 2i+3j=k B= -4i+2j-k The result by...

Okay. First off. This isn't homework. I have been working on a personal project and if I could get an equation for this it would really help me out. I'll try and explain the problem the best I can. What I need: An equation/method where I can specify an angle theta and a 3D line and it...

Gradient of a dot product identity proof? Homework Statement I have been given a E&M homework assignment to prove all the vector identities in the front cover of Griffith's E&M textbook. I have trouble proving: (1) ∇(A\bulletB) = A×(∇×B)+B×(∇×A)+(A\bullet∇)B+(B\bullet∇)A Homework...

Homework Statement so we have a (D) Line (geometry) it's Cartesian equation is ax+by+c=0 we have an A(\alpha,\beta) prove that the distance between the line(D) and the point A is d=\frac{la\alpha+b\beta+cl}{\sqrt{a^2+b^2}}Homework Equations The Attempt at a Solution let every distance be a...

$$ \frac{d}{dt}[\mathbf{a}\cdot (\mathbf{v}\times\mathbf{r})] = \dot{\mathbf{a}}\cdot (\mathbf{v}\times\mathbf{r}). $$ How is this true? Shouldn't the derivative affect the cross product as well?

The dot product A . B is the magnitude of vector A times the projection of B onto A. B . A is the magnitude of vector B times the projection of A onto B. Correct? A . B = B . A and this makes sense. But, say you're trying to find the components of a vector V in the direction of a vector W...

If you square the magnitude of a vector you get the dot product, correct? ||v||^2 = v . v Can you also say that ||v|| = sqrt(v . v)?

Homework Statement We consider two points, B and I and a line 'a'. B(0,-4,-7) I(-2,-2,-5) and a: x = y+1 = (z-2)/2 Determine the summits of A and C of triangle ABC knowing that: -Summit A belongs to the line 'a' -I is the foot of the height from A (perpendicular to BC) -The...

Hi, As I remember, dot product is commutative, and so (a.b).(a.b) = (a.a).(b.b) But when I apply to simple vectors it is all wrong, e.g: a = (2, 2, 0) b = (1, 0, 0) (a.b).(a.b) = 2.2 = 4 (a.a).(b.b) = 8.1 = 8 Why are they different? Pls explain for me Thanks

Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?

How would you show that the dot product between the normal unit vector of a plane and a position vector to any point on the plane is always the same without using this formula n.(r-r_0) = 0 ∴ n.r=n.r_0 where n is the normal vector, r and r_o are two position vectors to two points on the...

Hi everyone, I'm trying to understand the integral on http://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap14/chap14.pdf (page 14) I get all the steps except the how to get from eq. 48 to eq 49. I'm not really sure how to compute all the dot products. He let's the angle between n...