# Vectors Definition and 222 Discussions

In mathematics, physics and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a ray (a directed line segment), or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by

A
B

{\displaystyle {\overrightarrow {AB}}}
.A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.
Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.

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1. ### I Vectors as geometric objects and vectors as any mathematical objects

In geometry, a vector ##\vec{X}## in n-dimensions is something like this $$\vec{X} = \left( x_1, x_2, \cdots, x_n\right)$$ And it follows its own laws of arithmetic. In Linear Analysis, a polynomial ##p(x) = \sum_{I=1}^{n}a_n x^n ##, is a vector, along with all other mathematical objects of...
2. ### Momentum Collision Homework Problem -- help please

Parallel: M1V1+M2v2=M1V1’+M2V2’ (0.5)(3)+0=(0.5)(cos60)(3)+V2’Cos(x)(0.5) V2’cos(x)= Perpendicular: M1V1+M2v2=M1V1’+M2V2’ 0=(0.5)(0.3)(sin60)+V2’sin(x)(0.5) V2’sin(x)= And the divide 2 by 1 Which is tan(x)=2/1 And then plug then back in to solve, but I don’t think we do it like this because...
3. ### Vector equations for adding speeds while switching frames of reference in Galilean physics

I kind of just made up the questions. I realize this is a basic question but my knowledge of physics is very limited. q1 answer v_left_ball = v_left_ball - v_train v_right_ball = v_right_ball + v_train q2 answer To get the speed from Bob's frame I would use v_Bob = v_Bob + v_Alice To get...
4. ### I Why is momentum considered a vector and kinetic energy a scalar?

I'm not interested in the mathematical derivation, the mathematical derivation already is based on the assumption that momentum is a vector and kinetic energy is a scalar, thus it proves nothing. Specifically, what happens if we discuss scalarized momentum? What happens if we discuss vectorized...
5. ### 2D kinematics -- Calculate the acceleration of the jumping athlete

this is my work but the answers say 11 m/s^2 so I made an error somewhere. Also if someone could help me with solving the direction for the acceleration, that would be greatly appreciated.
6. ### Two forces acting on an object given in vectors - SOLVED

I tried splitting the forces up into F1 and F2 making Newtons second law equation into F1+F2=ma. Then I added over the the first force given. multiply the mass to the acceleration terms to get F2= (m*ai + m*aj) - F1
7. ### Where I can find similar examples to this kind of question? "given the electric field. E = (-3 / x*e0)i, find the charge density."

I'm preparing for exam but it seems I can't find problems similar to this on the internet. Here I will apply Gauss's law on the electric field vector to get the charge density. but the problem is that I can't find similar examples on the internet that uses direct vectors on Maxwell's equations...
8. ### I need help finding resultants using sine/cosine law

I need help finding the resultant with vectors: 37.5N[NE] and 45N[21° S of E] I just don't know a way to find the angles within this triangle to help me get the resultant, so can anybody help me out?
9. ### Dot product vectors problem

Summary:: I need to solve a problem for an assignment but just couldn't find the right approach. I fail to eliminate b or c to get only the magnitude of a. Let a, b and c be unit vectors such that a⋅b=1/4, b⋅c=1/7 and a⋅c=1/8. Evaluate (write in the exact form): - ||4a|| - 3a.5b - a.(b-c) -...
10. ### B Problem involving unit vectors

For example is this correct : 19icap.4(-i cap) = 76(i.-i)= 76 Or is it , take - out. Then -76(icap.icap)= -76 Is it -76 or 76 ?
11. ### Adding vectors in this 3-D problem

I gathered that the final position of the vectors when added up would be (M,-M,M), but I'm not sure if this is correct.
12. ### Find velocity with vector or without vector

At the moment he wrote that ##\frac{1}{2}mv_2^2=\frac{1}{2}m(-\dot{y}+\dot{x})^2## But, I know from vector ##v_2=\sqrt{(-\dot{y})^2+(\dot{x})^2}##. At first I (he) found that ##v_2=-\dot{y}+\dot{x}##. But, when thinking of simple velocity in ##x## and ##y## coordinate then I get...
13. ### Differential position vector

I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an...
14. ### Tricky conceptual Projectile motion question

So far all I have determined is the equations of motion for the two and that is as follows. It is trivial that y(t)=v1sin(Q)t -gt^2/2 and that x(t)=v2cos(Q)t. Now the angle that is anticlockwise from the negative horizontal of the robber is 90 - Q using basic trigonometry, using this we can...
15. ### I Trouble understanding contravariant transformations for vectors

Hey, so I've been studying some math on my own and I'm really confused by this one bit. I understand what contravariant components of a vector are, but I don't understand the ways in which they transform under a change of coordinate system. For instance, let's say we have two coordinate...
16. ### How to find d2 when given d1 and d, total time, and average velocity?

I rearranged the displacement formula to d2 = d + d1. I used cosine law to solve for d2 since the triangle is not right-angled but I am not getting the correct answer or angle for d2. The angle I used in cosine law (based on the diagram) was 32+12+90 = 134. d = v(t) = 130(3) = 390 km/h [N 32 E]...
17. ### Physics lab report calculating resultant forces -- help please

i have attached my attempt above i have sent it to my teacher and he said i should fix the mistakes and resend it
18. ### Calculate the dual basis and tangent basis vectors

a) Since ##tan(x/x_0)## is not defined for ##x=\pm\pi/2\cdot x_0## I assume x must be in between those values therefore ##-\pi/2\cdot x_0 < x < \pi/2\cdot x_0## and y can be any real number. Is this the correct answer on a)? b) I can solve x and y for s and t which gives me ##y=y_0\cdot s## and...
19. ### I Scalar product of biharmonic friction with velocity components

I know that taking the scalar product of the harmonic (Laplacian) friction term with ##\underline u## is $$\underline u \cdot [\nabla \cdot(A\nabla \underline u)] = \nabla \cdot (\underline u A \nabla \underline u) - A (\nabla \underline u )^2$$ where ##\underline u = (u,v)## and ##A## is a...
20. ### I Principal difference between complex numbers and 2D vectors revisited

I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question. I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...
21. ### Translational and rotational velocity

For a cylinder rolling down an inclined plane, does the tangential velocity of a point a distance R from the axis of rotation equal the velocity of the center of mass?
22. ### Converting Velocity Vector Formula from Cartesian Coordinate System to Polar Coordinate System

I have a little question about converting Velocity formula that is derived as, ##\vec{V}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}+\frac{dz}{dt}\hat{z}## in Cartesian Coordinate Systems ##(x, y, z)##. I want to convert this into Polar Coordinate System ##(r, \theta)##...
23. ### I Div Grad Curl question

I'm reading div grad curl for my math methods class, and I came across this question: "Using arrows of the proper magnitude and direction, sketch each of the following vector functions: (a) iy + jx, (b) (i + j)/√2 I don't understand the notation. Why is there an y and x next to the i and j in...
24. ### Stuck on a few Vector homework problems

I'm stuck on a few Vector homework problems. I don't quite understand how to write vectors A+B and A-B for questions 1b and 2b. I tried starting with calculating the magnitude for vector A+B on question 1b and then followed by finding theta, but I'm not sure if that's what I'm supposed to do...
25. ### Vectors physics homework -- Addition of two vector forces

Trying to solve it
26. ### Vectors Homework

Can’t seems to find the degree
27. ### I Why is there a contradiction?

Let: ##\nabla## denote dell operator with respect to field coordinate (origin) ##\nabla'## denote dell operator with respect to source coordinates The electric field at origin due to an electric dipole distribution in volume ##V## having boundary ##S## is: \begin{align} \int_V...
28. ### How do I show that 2 moving objects collide?

They collide when their positions are the same, so I plugged the information for the boat into equation 1 to get an expression for d which is (2i, +j)t^2 Then I used equation 4 to get an expression for d for the branch, which is (-4i, +j)t I would need to take into account the different...
29. ### Relative Velocity Of Swimmer

So I was just wondering if someone could check my method for (b) as sometimes I can have a tendency of getting the relative components wrong ect. Diagram 1 (a) Time for PY: ##T=L/c## Time for YP: ##T=L/c## Total Time:##2L/c## (b) Velocity for PY: ##c-v##...
30. ### Find the incorrect relation from the figure

I tried making B vector in direction of D vector with a minus sign and after doing so I got the answer C vector - D vector= -A vector. But it's given as incorrect. I don't know why. Please explain how other options are correct.
31. ### I Confusion about index notation and operations of GR

Hello, I am an undergrad currently trying to understand General Relativity. I am reading Sean Carroll's Spacetime and Geometry and I understand the physics (to a certain degree) but I am having trouble understanding the notation used as well as the ideas for tensors, dual vectors and the...
32. ### Intuition behind dot product

I know that a dot product of 2, 2 dimension vectors a, b = (ax * bx) + (ay * by) but it also is equal to a*bCos(θ) because of "projections". That we are multiplying a vector by the 'scalar' property of the other vector which confuses me because that projection is in the direction of the...
33. ### I Vector math (small angle approximation)

Given the following vectors: how can i determine that Θ = Δp/p ? I can understand that p + Δp = p' but nothing arrives from this. Any help is welcome!
34. ### Calculating a force on a member (Statics)

I isolated the member ABC and drew the free body diagram: α is then calculated using inverse tan: Tan-1=(6.25+15)/50=23.03 Then force of member BD on the joint can be found by sum of all moments around point A. Then Ax is calculated which is equal to BD×Cos(α)=235.2×Cos(23.03) Ax=216.48...
35. ### Vectors and vector addition

Homework Statement Calculate |u+v+w|, knowing that u, v, and w are vectors in space such that |u|=√2, |v|=√3, u is perpendicular to v, w=u×v. Homework Equations |w|=|u×v|=|u|*|v|*sinΘ The Attempt at a Solution [/B] Θ=90° |w|=(√2)*(√3)*sin(90°)=√(6) Then I tried to use u={√2,0,0}...
36. ### Vectors and scalar projections

Homework Statement Let a and b be non-zero vectors in space. Determine comp a (a × b). Homework Equations comp a (b) = (a ⋅ b)/|a| The Attempt at a Solution [/B] comp a (a × b) = a ⋅ (a × b)/|a| = (a × a) ⋅b/|a| = 0 ⋅ b/|a| = 0 Is this the answer? Or is there more to it?
37. ### Finding the expression for the x-component of velocity (vectors?)

Homework Statement a particle's position is the vector r=(ct^2-2dt^3)i+(4ct^2-dt^3)j where c and d are positive constants. find the expression for the x-component of the velocity (for time t>0) when the particle is moving in the x-direction. you should express your answer in terms of variables...
38. ### Finding an area of a triangle formed by three points

Homework Statement P(3, 0, 3), Q(−2, 1, 5), R(6, 2, 7) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR Homework Equations A = \frac{1}{2}|\vec{AB}\times\vec{AC}| Source...
39. ### Courses Is it okay to not understand the calculus in intro physics?

I don't understand a good portion of the non-algebraic math behind much of the physics in my first semester college class. I understand everything with algebra, and can solve all problems, but I don't understand the relationships with vector cross/dot products, calculus derivations, DE, etc...
40. ### B The definition of “vector” in math and physics

I'm learning APL and this is how a vector is defined https://tryapl.org: All data resides in arrays. An array is a rectangular collection of numbers, characters and arrays, arranged along zero or more axes. We can use more specific terms for some arrays, like a single number is a scalar, a list...
41. ### How do you find the acceleration needed to clear a jump?

Homework Statement Homework Equations The Attempt at a Solution I used Pythagorean theorem to find the length of the ramp (25^2+18^2 = √949) and found the angle of elevation using tangent (Tanθ=18/25) but then got stuck on what formula to use.
42. ### Vector Addition Question: find angle (A+B & A-B)

Homework Statement Two vectors A and B have precisely equal magnitudes. For the magnitude of A + B to be 65 times greater than the magnitude of A - B, what must be the angle between them? Homework Equations The Attempt at a Solution I tried using the dot product and solving for the angle but...
43. ### B Directions of Vectors

What's the difference between saying that a vector's direction is north of east and north east?
44. ### Defining which cyclist profits the most from slipstream

Homework Statement - Question: Which cyclist (A, B or C) profits the most from cyclist slipstream (also called “aerodynamic drafting”)? - Given: the direction of the wind, the positions of each cyclist; an illustration representing this - NOTE that I am expected to solve this question...
45. ### Vector torque problem: Force applied to a disc

Homework Statement A force of magnitude 50N is applied at the bottom point Q of a disk of radius 8m that is pinned at P (leftmost point) See attached picture (a) Find the angle between the force and the vector from P to Q. (b) Find the magnitude of the applied torque. Homework Equations...
46. ### Extending an aircraft landing gear that has gotten stuck

Imagine an aircraft of which the its wing is connected to its landing gear by means of a hinge (joint), and then the gear suddenly gets stuck midway. If, subsequently, the aircraft makes a turn around its longitudinal axis (see picture), it seems to hold true that then the angle between the...
47. ### B Effect of angle on force

Hi, If 2 people are holding a bag at an angle of 45 degrees each, and then only person is going to hold it, it is being said that the force that will have to be applied by that one person will be 1.5 times greater than when he was applying it together with the other. Can anyone explain this or...
48. ### Resolution of vectors for problems related to mechanics

Homework Statement Refer the given image. [Prob 2.9] Homework Equations F=ma The Attempt at a Solution I drew the normal vector perpendicular to the surface of the cone and resolved it as ##Nsin\theta=mg## ##Ncos\theta=\frac {mv_{0}^2} {r}## where ##v_{0}## and ##r## are the speed and radius...
49. ### B Addition of Perpendicular Vectors in two ways

I have a Force Vector = 100N, making an Angle = 45 degrees with x-axis. When I find their Components trigonometrically, I get 70N each; as Fx = 100xCos(45) = 70N Fy = 100xSin(45) = 70N Verifying the result, by Head-to-Tail method, I get 70N + 70N = 140N. Why is there discrepancy or where am I...
50. ### Direction of angular velocity

Angular velocity is the rate of angular displacement about an axis. Its direction is determined by right hand rule. According to right hand rule, if you hold the axis with your right hand and rotate the fingers in the direction of motion of the rotating body then thumb will point the direction...