What is Vector addition: Definition and 183 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


{\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. P

    MATLAB Animating Vector Addition in Matlab (Looking for improvements)

    Here is my code thus far and it seems to work (I've attached a gif as well so hopefully that works) va1 = [50*cos(pi/3), 50*sin(pi/3)]; %Two different vectors both starting from the origin vb1 = [20*cos(pi/6), 20*sin(pi/6)]; iterator = linspace(1,100,100); %100 iterations for the animation...
  2. kai sinclair

    Efficient Calculation of Final Magnitude and Angle in Vector Addition Method

    okay so I'm a Electrician I've found short method of calculating the final magnitude of a system (Lₜ), this relies mainly on Eucliud's axiom of angles within parrallel lines and is this ∑(cos(θₙ-θₜ)⋅Lₙ)=Lₜ where Lₙ and θₙ are the initail manignitude and angles respectively, and θₜ is the final...
  3. Slimy0233

    Vector addition and subtraction

    Mentor note: Moved from General Math section, so is missing the homework template. If you want the vector which represents the base from 1 to 2 (or O1 to O2 are the position vectors, I am using co-ordinates as names), what would O12 (vector from 1 to 2) and why? 1. Would it be O1 - O2 or O1 +...
  4. susan_khan

    Airplane Vector Problem: Drawing the diagram

    An airplane has an air velocity of 500 km/h [N 30 E] and encounters a wind from [S 75 W] at 180 km/h, find the ground velocity. Make sure you draw a big, labelled diagram. Please help! I’m understand the calculations that need to be done (cosine law then sine law for the angle) but I’m a little...
  5. E

    Velocity vector addition problem

    Hello, guys. Interesting riddle here. I have no idea how to solve it. Tried different methods, but point is answer is always wrong, exact answer Downriver, at an angle of 53.13(degree) to the bank. That exercise is from "Pohl’s Introduction to Physics"
  6. guyvsdcsniper

    Where am I going wrong with this vector addition?

    I am following along with an example in my book regarding force from an electric charge. I understand the process but I believe I am getting something wrong when it comes to adding the vectors. Essentially, F13 is equal to -1.35*10^-3 j and when I add that to the j component of F23 which is...
  7. brotherbobby

    Swimming to and fro across a river

    Problem statement : While the statement is given above, I draw the given image for the problem. Attempt : It is clear at the outset that, since the velocity of river is constant, the man will not take the same time for the forward and reverse journeys. For both journeys however, the Relevant...
  8. G

    To find the relative velocity of the wind as seen from a car

    Could I please ask for any help with the following question: Here's my attempt: (i and j are unit vectors in the directions of east and north respectively) (apologies that LaTeX is simply not working for me, I'll label the angles in each case T and P as shown in my diagram) Let the...
  9. bagasme

    B Derivation of Cosine and Sine Method of Vector Sum

    Hello all, In high school physics, the magnitude sum of vector addition can be found by cosine rule: $$\vec {R^2} = \vec {F^2_1} + \vec {F^2_2} + 2 \cdot \vec F_1 \cdot \vec F_2 \cdot cos ~ \alpha$$ and its angle are calculated by sine rule: $$\frac {\vec R} {sin ~ \alpha} = \frac {\vec F_1}...
  10. K

    I Vector addition in spin orbit coupling

    Hello! I am reading about spin-orbit coupling in Griffiths book, and at a point he shows an image (section 6.4.1) of the vectors L and S coupled together to give J (figure 6.10) and he says that L and S precess rapidly around J. I am not totally sure I understand this. I know that in the...
  11. jonander

    I General equation for the magnitude of the difference vector

    Hi everyone, While finding the solution for one of my exercises, I found the following answer. I'm seriously questioning if the equations provided in that answer are reversed. According to my understanding, if two vectors ##\vec{S}## and ##\vec{T}## are parallel (same direction) the magnitude...
  12. akashpandey

    I Triangle law of vector addition and the Pythagoras theorem

    i know its pretty basic but please give some insight for triangle law of vector addition and pythgoras theorem. becuase ofcourse if you use traingle law to find resultant it will be different from what is pythagoras theorem
  13. JD_PM

    Arguing about the magnetic force vector

    I am writing about the nature of force in classical mechanics and what does really imply, in terms of change in motion. I am using as an example a circuit, on which we exert a force. I am trying to justify the following scheme (concretely, ##f_{mag}##): The thing is that I am wondering how...
  14. megaly

    Slope Problem: Need Help with a Difficult Equation?

    Im really stuck with this problem ! any help would be much appreciated
  15. hnnhcmmngs

    But, as I said, you don't actually need the coordinates at all.

    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}...
  16. Charge

    Adding Vectors to Find Velocity

    Problem: Dr. L and his cat Kepler are coming home from fishing. They ended their trip on the south bank of a river, directly across a 1200 m wide river, from where Caroline was going to pick them up on the north bank. They are in identical boats that can travel at 4 m/s (Vb). The river is...
  17. opus

    B Vector addition- Positive x axis.

    Please see the attached page to see what I'm talking about. In the top right paragraph, it states to use "the positive direction of the x-axis". It is given that ##θ_2=30°## and it shown visually at the bottom of the page. In the problem it's using -60° and I'm not sure how they're getting that.
  18. L

    Effective acceleration due to gravity in non-inertial frame

    Take some sort of system accelerating with respect to an inertial reference frame: let's take a spherical mass on the end of a string forming a simple pendulum with the ceiling of a car, and allow that car to accelerate uniformly. Could someone share with me how they interpret the concept of a...
  19. Specter

    Proving the associative property of vector addition

    Homework Statement Give an example of the associative property of vector addition using vectors in Cartesion form. Homework Equations (u+v)+w=u+(v+w) The Attempt at a Solution I can't figure out how to get the arrow on top of my work so I wrote it without it. I'm somewhat confused on why I...
  20. C

    Simple Vector Addition Problem

    Homework Statement "A bird flew 40 m to the west, then 100 m in a direction 36.9 degrees to the north of east. Use the algebraic addition of vectors to fins the magnitude of the bird's net displacement" Use sin (36.9) =0.6 and cos (36.9) = 0.8 Homework Equations Vector addition? A =...
  21. alexi_b

    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 EquationsThe Attempt at a Solution I tried using the dot product and solving for the angle but i...
  22. B

    Force Resultants: Comparing Rules

    I just want to know the difference between those rules: 1. R^2 = F1^2 * F2^2 + 2*F1*F2*COS(the angle between F1 and F2) 2. The second is about the parallelogram rule, it says that the two vectors are added and their summation is the magnitude of the resultant. Which one is correct?
  23. baldbrain

    River-boat problem (Relative velocity)

    Homework Statement A man wishes to cross a river of width 120 m by a motorboat. His rowing speed in still water (or relative to water) is 3 m/s and his maximum walking speed is 1 m/s. The river flows with a velocity of 4 m/s. (a) Find the path which he should take to reach the point directly...
  24. Jan Berkhout

    How do I do this trigonometry vector calculation?

    Homework Statement A pilot wishes to fly at maximum speed due north. The plane can fly at 100km/h in still air. A 30km/h wind blows from the south-east. Calculate: a) The direction the plane must head to fly north. b) Its speed relative to the ground. Homework Equations Sine Rule...
  25. xaratustra

    Vector addition and Newton's law

    I know that n-body problem can be complicated, but that's for the dynamics. What about a static case: e.g. if I have the distances of several bodies A, B and C etc. and their distance to a reference mass m, can I just use the vector addition of the Newton's gravitational force to add up all of...
  26. F

    Find All Possible Values of x When 3 Vectors are Linearly Dependent

    Homework Statement >There are three vector $$ \vec a ,\vec b, \vec c$$ in three-dimensional real vector space, and the inner product between them $$\vec a . \vec a=\vec b.\vec b=\vec a.\vec c=1, \vec a.\vec b=0, \vec c.\vec c=4 $$ When setting $$x = \vec b.\vec c$$ , (dot here means dot...
  27. H

    Problem using vectors and relative motion

    Homework Statement A river of width 4 km flows at 5 km/h. A swimmer whose speed relative to water is 4 km/h, starts swimming from a point A on a bank. What minimum distance will the swimmer have to walk on the other bank to reach point B directly opposite to A? [/B]Homework Equations Vba = Vb...
  28. V

    Vector addition and the force applied to the shaft of the pulley

    Homework Statement Hello, I am currently studying mathematics and physics on my own, and I ran into a type of problem in physics that is a bit unclear to me. Say we have pulley at the top of an inclined plane with a base angle of alpha and the top angle is alpha. Homework Equations I assume...
  29. Sandro Romualdez

    Vectors: Proving the value of a•(b x c)

    Homework Statement Given that vector a = (1, 2, -5), b = (-12, 41, 75) and c = a + 2b, explain why (without doing any calculations whatsoever) the value of a•(b x c) = 0 Homework Equations No specific equations, as the question asks for the value without making any calculations. This problem...
  30. Physics345

    Sketching resultants using vector addition.

    Homework Statement Given the following diagram, use vector addition to sketch the resultants: a) u→−2v→ b) 3v→−u→ Homework EquationsThe Attempt at a Solution For this question I am fairly certain that it is correct but for some reason I'm doubting myself. I was wondering if I could get some...
  31. isukatphysics69

    General question about vector addition -- using graphical versus mathematical methods

    Homework Statement I have an exam on vectors and 2 dimensional motion today. I NEED to know if it is possible to solve any GRAPHICAL VECTOR PROBLEM using only INTEGRATION AND MATH WITHOUT ADDING VECTORS TOGETHER GRAPHICALLY. So given any word problem that is supposed to be solved using that...
  32. V

    Vector problem -- Find the angle between vector A and vector B

    Homework Statement Vector A + Vector B + Vector C = 0 A=B+C Homework Equations Find the angle between vector A and vector B The Attempt at a Solution I tried to solve this problem by resolving all the vectors and individually equating the value of vectors at each axes x, y and z to 0 but got...
  33. S

    How do you graph vectors on a graph paper using their magnitude and direction?

    Homework Statement Vector A: 1.96N at 20° Vector B: 1.71N at 65° Homework EquationsThe Attempt at a Solution I change the Newtons to cm. Vector A: 1.96cm and Vector B: 1.71cm. I multiplied it by 2 so it would look bigger on the graph paper. Vector A: 3.92cm and Vector B:3.42cm. I was able to...
  34. W

    Methods in Calculating with Vector Components

    <Moderator's note: Moved from a technical forum and therefore no template.> Homework Statement A proton (q = 1.60 x 10-19) is in a uniform, 0.500 T magnetic field. This proton has velocity components vx = 1.50 x 105, vy = 0, and vz = 2.00 x 105 m/s. Find the force on the proton at t=0. 2...
  35. andylatham82

    What Is the Correct Calculation for the Running Dog Vector Problem?

    I've been racking my brain for a while now over what I feel should be a simple problem to solve, but my answer is wrong. I'm not wildly wrong, but wrong enough to think I've made a proper error and it's not just a rounding error or something. The question goes like this: A dog in an open field...
  36. Alettix

    Point inside a tetrahedron with vectors

    Homework Statement As part of a longer problem: "Find necessary and sufficient conditions for the point with positionvector r to lie inside, or on, the tetrahedron formed by the vertices 0, a, b and c." Homework Equations I am not sure... vector addtion? The Attempt at a Solution I don't...
  37. MickeyBlue

    Relative velocity of a ball on a train

    Homework Statement A train is traveling east along a straight run of track at 72.0 km/hr. Inside, two siblings 1.9 m apart are playing catch directly across the aisle. The kid wearing a P.J.Harvey T-shirt throws the ball horizontally north. The ball crosses the train and is caught 0.75 s later...
  38. CynicusRex

    B Mistake when explaining associativity of vector addition

    Does he make a mistake at 6:18? In case of associativity. When doing a+(b+c) he's just placing the tip of a to the tail of (b+c), but shouldn't he have added the tail of a to the tip of (b+c) ending in a different point? I understand vector addition is associative, but I think how he did it is...
  39. parshyaa

    I Parallelogram law of vector addition

    Acording to this diagram vector P = vector BC and vector Q = vector OB(their magnitudes are also respectively equal.) Therefore acoording to the congruency of triangles angle alpha = (theta)/2. But this is not right( what's wrong ) {Tan(alpha) = Qsin(theta)/(P + Qcos(thets)) ] Why resultant...
  40. C

    Momentum, vector addition question

    Homework Statement A car with a mass of 1400 kg is westbound at 50 km/h. It collides at an intersection with a northbound truck having a mass of 2000 kg and traveling at 40 km/h. What is the initial common velocity of the car and trucks after collision if it's an inelastic collision? Homework...
  41. Amine_prince

    B Error in Vector Addition: U & V Perpendicular

    i have two unitary vectors in space U and V , U and V are perpendicular . W is a vector that verifies W ^ V = U - W . the following resolution is incorrect , i wan't to understand why : we use (o,U,V,(U^V)) . components of U (1,0,0) , V(0,1,0) , W(a,b,c) where a , b and c are real...
  42. J

    What is the resultant of the vector from this impact?

    Homework Statement two people kick a ball. one of them kicks it with a force of 215.0 Newtons. the other one with 189.0 Newtons. their approach creates a 210.0 degree angle at the point of impact and causes the ball to go in a different direction. what is the resultant of the vector from this...
  43. clydefrog

    Plane ride relative motion problem

    Homework Statement You are traveling on an airplane. The velocity of the plane with respect to the air is 160 m/s due east. The velocity of the air with respect to the ground is 41 m/s at an angle of 30° west of due north. 1) What is the speed of the plane with respect to the ground? Homework...
  44. J

    How Do You Sum the Components of Vectors in Three Dimensions?

    Homework Statement [/B] Given the following vector components of vectors C and D: Cx = 8.10, Cy = -5.40, Cz = -7.90, Dx = 4.40, Dy = -2.50, Dz = 4.50, find the x, y, z components of their sum. Homework Equations No relevant equations that I know of...[/B] The Attempt at a Solution I am...
  45. D

    What Angle Should the Goose Fly to Head Directly Southward?

    Homework Statement Canada geese migrate essentially along a north-south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 km/h. The one goose is flying at 100 km/h relative to the air but a 38 km/h -km/h wind is blowing from west to east. At what...
  46. J

    Help with Vector Addition Problem

    Homework Statement In photo: https://gyazo.com/6348b5ffe368852ee9868a2ba0dd7e38 Homework Equations In image The Attempt at a Solution In the boxes Where did I go wrong with the angle calculation? Thanks
  47. Blockade

    How do I know that I should do a vector addition for this?

    I don't need help finding the resultant of this example problem. What I do need help for is: What exactly from this prompt hinted that I must add all the vectors provided. I have read this many times through and I still didn't know what to do. How does adding the vectors help them find the...
  48. Mr Davis 97

    Why is vector addition defined the way it is?

    We all know that arithmetic addition arose not out of some axiomatic system, but out of the natural tendency to combine similar objects. I am satisfied with typical addition being loosely defined in this way. But when it comes to addition with other objects, like vectors, I am little bit...
  49. M

    Vectors given the resultant and angle

    Homework Statement The figure depicts the sum of two velocities, vA and vB. The value of the sum is7 m/s and the angles shown in the image are θ1 = 29.8°, and θ2 = 18.3°. Homework EquationsThe Attempt at a Solution I have no idea where to even begin on this