Dot Definition and 564 Threads

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...

Homework Statement The point P and Q have postion vectors a + b, and 3a - 2b respectively, relative to the origin O.Given that OPQR is a parallelogram express the vector PQ and PR in terms of a and b. By evaluating two scalar products show that if OPQR is a square then |a |2 = 2 |b |2 The...

Homework Statement F(x,y)=<2xy,x^2+y^2>, the path is part of the unit circle in the 1st quadrant. And I'm supposed to calculate ∫F°ds given that info Homework Equations My question is if this equation would apply to the following problem ∫F°ds=θ2-θ1 Since this is a circular equation.

Homework Statement Calculate F=∇V, where V(x,y,z)= xye^z, and computer ∫F"dot"ds, where A)C is any curve from (1,1,0) to (3,e-1) B)C is a the boundary of the square 0≤x≤1, 0≤y≤1... oriented counterclockwise. Homework Equations ∫F"dot"ds= ∫F(c(t)"dot"c'(t) The Attempt at a Solution...

I'm wondering if anyone can help me with the reasoning for what follows, I'm thinking I must just be missing something quite obvious but I can't seem to see what that is at the moment! :redface: I understand how to get the structure by starting from a 'skeleton structure' and just filling in...

Homework Statement #1 Given that the angle between the vectors a and b is 2Pi/3 and |a|=3 and |b|=4 calculate: (axb)^2 [(2a+b)x(a+2b)]^2 #2 Given three unit vectors, a, b, c such that a+b+c=0 find (a dot b) + (b dot c) + (c dot a) #3 Given AB=a+2b BC=-4a-b CD= -5a-3b...

Simple question, but I don't know why I never learned this before. If the scalar projection of vector B onto vector A is B * Unit vector of A (or [A dot B]/[magnitude of A]), then what does the dot product of simply A and B give you, assuming neither is a unit vector. If it's not clear what...

Hello Forum, When we represent a vector X using an orthonormal basis, we express X as a linear combination of the basis vectors: x= a1 v1 + a2 v2 + a3 v3+ ... Each coefficient a_i is the dot product between x and each basis vector v_i. If the vector x is not a row (or column vector)...

Hey guys I am a beginner in linear algebra. I am doing vectors now and I just noticed that when two vectors are parallel (or antiparallel), the product of their norms is equal to the absolute value of their dot product, or |u \cdot v | = ||u|| \ ||v|| I know that this is a special case of...

Homework Statement Homework Equations Tan θ=A.B/|A|*|B| The Attempt at a Solution A=4i-9j B=9i-6j A.B=-18 Magnitude of A = √97 Magnitude of B = √117 Solve that out using Cos-1 (-18/Sqr97*Sqr117) and I keep getting the wrong answer.

I'm just learning this Latex(sic) formatting, so it's not ideal. I was trying to explore the geometrical significance of the cross product when I happened upon an interesting observation. I've seen things like this before, but never had time to really examine them. I define two vectors...

Another vector identity I have been trying to prove. My textbook lists this identity in "properties of cross products" without proving it. I have been trying to prove it, withou much luck, so some insight is appreciated. u \times (v \times w) = (u \cdot w)v - (u \cdot v)w Thanks! BiP

Could anyone give me a link that explains or simply explain to me how the dot convention on the mutual inductance works? For example, I'm trying to draw an equivalent circuit that converts the dots into voltage sources of this circuit...

Hi everyone, I'm working with quantum dot (PbS). However, I'm actually not a chemist, so could anyone help me to address some issues regarding quantum dots especially related to ligand exchange process. My questions are: 1. anyone can address about what kind of functionality (e.g: thiol...

the calculation of many quantities involve cross and dot products in their formulae. For cross products, i only know of three: lorentz force, angular momenta and torque. For dot products, i only know of one: work done im hoping to add more to my list. Can you guys help me include other...

Question on Molecular Orbital Theory (and the "dot" that represents no interaction) I have a question about the "dot" that represents no interaction between orbitals. For example, in the molecular orbitals of H3, there is the lowest energy molecular orbital that has two bonding interactions...

Homework Statement the question is simply asking to prove that if speed is constant than velocity and acceleration vectors are perpendicular to each other. it also says as hint to differentiate v dot v= u^2... V=velocity A=acceleration u=speed t=time Homework Equations i suppose...

Hey guys, this is for my classical E&M class but it's more of a math problem. Homework Statement Show: ∇(\vec{A} . \vec{B}) = \vec{B} \times (∇ \times \vec{A}) + (\vec{B} \times ∇)\vec{A} + \vec{A} \times (∇ \times \vec{B}) + (\vec{A} \times ∇)\vec{B} Homework Equations I tried...

This is something that has been bothering me... Given two vectors A and B Is there a way to prove that A dot B = ABcosθ ? I'm concerned with WHY this is the case... If anyone has a good proof that would be great.

Homework Statement Show that the vector fields A = ar(sin2θ)/r2+2aθ(sinθ)/r2 and B = rcosθar+raθ are everywhere parallel to each other. Homework Equations \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(0) The Attempt at a Solution So, if the dot product equals 1. They should be...

Homework Statement Given the nonzero vector a ε ℝ3, a\dot{}x = b ε ℝ, and a × x = c ε ℝ3, can you determine the vector x ε ℝ3? If so, give a geometric construction for x. Homework Equations a\dot{}x = ||a||||x||cos\Theta The Attempt at a Solution I'm not really certain what it is...

Homework Statement 1. Suppose that u + v + w = 0. Show that u x v = v x w = w x u. What is the geometric interpretation of this result? (Note: The interpretation should explain both the length and the direction). 2. Let v1, v2, and v be three mutually orthogonal vectors in space. Use the...

Homework Statement The two vectors a and b lie in the xy plane and make angles α and β with the x axis. a)By evaluating a • b in two ways (Namely a •b = abcos(θ) and a • b = a1b1+a2b2) prove the well-known trig identity cos(α-β)=cos(α)cos(β)+sin(α)sin(β) b)By similarly evaluating...

Question If a = i - 2j + 2k b = 4i - 3j M = 3ii + 2ij - ji + kj. M is a dyadic. Determine a\bullet2M\bulletb Homework Equations The Attempt at a Solution This isn't really homework, but rather just some self study to help me understand some of my work better. Attached is the...

Homework Statement The proof begins: Suppose that F is conservative. Then a scalar field ε(r) can be defined as the line integral of F from the origin to the point r. So ∫F dot dr = ε(r), where the limits of integration are from 0 to r. The next step, however, eludes me: From the...

Homework Statement r(t)=(x(t),y(t),z(t)) t has been chosen so that r'.r'=1 show that r'.r''=0 Homework Equations v'.w'=|v||w|cos(theta) The Attempt at a Solution Clearly what is being described is circular motion about a unit circle. And using the equation for a unit circle its easy to...

Just wondering... If the dot product of the acceleration and velocity vectors is zero, then does v2/r = 0 have to be true? If this is true, is it possible to prove it? If the statement is false, is it possible to prove that as well?

greetings, consider two vector as it is A(1,0) and B(-1,0). now if we find the projection of A on B we should get zero but its coming -1.where i am going wrong? advanced thanks.

Homework Statement The following species has been discovered in interstellar space: HNC. Draw the Lewis Dot diagram. Homework Equations N/A The Attempt at a Solution I tried several different configurations (see below), but they were all marked incorrect. Here is the basic method...

There is mention that dot products of vectors are used in backface removal in 3d graphics. Does anyone know of any article which explains the application of dot products to this area of graphics? I am looking for a concrete example.

Homework Statement The Attempt at a Solution do you see where it sees sqrt((1-i)(1+i)+9)? It should be (1+i)(1+i) Why isn't it?

Homework Statement A dot B=0.707m^2, A cross B=4.950m^2 k^. If |A|=2.500m and B makes an angle of 135° with the positive x-axis, what are A and B in component form? Homework Equations A*B = |A||B|cos(θ) A X B = |A||B|sin(θ)@RHR The Attempt at a Solution I have no prior physics...

While we calculate cross product of two vectors let A and B we write ABsinθ. And while we calculate dot product of them we write ABcosθ. Why particularly we use sinθ for cross product and cosθ for dot product.Is there any physical reason why we choose sine for cross product and cosine for...

Why is dot product given by a*b cosθ whereas cross product ab sinθ.

What is the difference between matrix multiplication and the dot product of two matrices? Is there a difference? If, A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and B = \begin{pmatrix} e & f \\ g & h \end{pmatrix} then does {\mathbf{A} \cdot...

Hello buddies, Here is my question. It seems simple but at the same time does not seem to have an obvious answer to me. Given that you have two vectors \mathbf{u},\mathbf{v}. They are orthogonal \mathbf{u}^T\mathbf{v}=0 by standard dot product definition. They have norm one...

Homework Statement Find the cosines of the internal angles of the triangle which has the following coordinate vectors as its vertices: A(4,0,2) B(6,2,1) C(5,1,6) Homework Equations I understand that you have to find the 3 vectors which represents each side of the triangle in order to...

let y1=(1,0,i,0) and y2=(0,i,1,0) what is (y1.y2)/(y1.y1)? I make it i/(1+i2) = i/0 which seems incorrect. My notes seem to give the answer as -i/2 and I don't understand how this was calculated. Any help is appreciated. Thanks

i m really confused..please explain with a physical example so that I can learn the application of it

Homework Statement I have vector R. I need to show the R dot dR/dt = 0 => 1/2 d/dt[R dot R] Homework Equations The Attempt at a Solution I guess I've never really applied the chain rule to dot products and its throwing me off. How does one go from R.dR/dt=0 to 1/2 d/dt[R.R] = 0. I...

Hi there. I have this problem, which says: In the cartesian system the tensor T, twice covariant has as components the elements of the matrix: \begin{bmatrix}{1}&{0}&{2}\\{3}&{4}&{1}\\{1}&{3}&{4}\end{bmatrix} If A=e_1+2e_2+3e_3 find the inner product between both tensors. Indicate the type and...

What does this mean? (\vec A\cdot\nabla)\vec B I read somewhere that (\vec A\cdot\nabla)\vec B=\vec A\cdot\nabla\vec B but this must be nonsense since you can't take the gradient of a vector.

Reason I posted this in the maths help forum is that an equation of this form randomly popped up in a homework I was doing on differential geometry. I started with a one-form ω=dβ (β is a scalar function) and found that if for a random vector v, ω(v) = 0, then \frac{d}{dt} \left(...

Homework Statement Let c(t) be a path of class C1. Suppose that ||c(t)||>0 for all t. Show that c(t) dot product with d/dt((c(t))/||c(t)||) =0 for every t. Homework Equations I am having trouble with the derivative of (c(t)/||c(t)||) and how to show that when its dotted with c(t) that...

Explain, for example, why you can cross three vectors (two at a time, following the usual rules), but not dot three vectors. Do you see the dot product "in action" in matrix multiplication? What sort of insights can the dot product give when trying to comprehend matrix multiplication?

Homework Statement u (dot) [ u x v] Homework Equations The Attempt at a Solution The answer is 0, but I'm not sure why. Do you simplify [u x v], then use that vector to dot it with u? Or can you distribute the u to both u and v?

Homework Statement Let 'u' and 'v' be two non zero vectors such that the prjection of 'u' along 'v' equals the projection of 'v' along 'u.' Using the formula for projection, show that 'u' and 'v' are either perpendicular or parallel. Homework Equations The Attempt at a Solution...