What is Angle between vectors: Definition and 38 Discussions

Cosine similarity is a measure of similarity between two non-zero vectors of an inner product space. It is defined to equal the cosine of the angle between them, which is also the same as the inner product of the same vectors normalized to both have length 1. The cosine of 0° is 1, and it is less than 1 for any angle in the interval (0, π] radians. It is thus a judgment of orientation and not magnitude: two vectors with the same orientation have a cosine similarity of 1, two vectors oriented at 90° relative to each other have a similarity of 0, and two vectors diametrically opposed have a similarity of -1, independent of their magnitude. The cosine similarity is particularly used in positive space, where the outcome is neatly bounded in


{\displaystyle [0,1]}
. The name derives from the term "direction cosine": in this case, unit vectors are maximally "similar" if they're parallel and maximally "dissimilar" if they're orthogonal (perpendicular). This is analogous to the cosine, which is unity (maximum value) when the segments subtend a zero angle and zero (uncorrelated) when the segments are perpendicular.
These bounds apply for any number of dimensions, and the cosine similarity is most commonly used in high-dimensional positive spaces. For example, in information retrieval and text mining, each term is notionally assigned a different dimension and a document is characterised by a vector where the value in each dimension corresponds to the number of times the term appears in the document. Cosine similarity then gives a useful measure of how similar two documents are likely to be in terms of their subject matter.The technique is also used to measure cohesion within clusters in the field of data mining.The term cosine distance is often used for the complement in positive space, that is:







{\displaystyle D_{C}(A,B)=1-S_{C}(A,B),}



{\displaystyle D_{C}}
is the cosine distance and



{\displaystyle S_{C}}
is the cosine similarity. It is important to note, however, that this is not a proper distance metric as it does not have the triangle inequality property—or, more formally, the Schwarz inequality—and it violates the coincidence axiom; to repair the triangle inequality property while maintaining the same ordering, it is necessary to convert to angular distance (see below).
One advantage of cosine similarity is its low-complexity, especially for sparse vectors: only the non-zero dimensions need to be considered.
Other names of cosine similarity are Orchini similarity and the Tucker coefficient of congruence; Ochiai similarity (see below) is cosine similarity applied to binary data.

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  1. brotherbobby

    A moving boat with a flag on its mast

    Let me copy and paste the problem as it appeared in the text. Attempt : First off, the launch can have no velocity of its "own" relative to air current. It is moving entirely due to the air current. Moreover, since no values are given for river, the river current velocity can't be found...
  2. S

    B Multiple Rotating Signs on a Fence Gate -- Overlap Question

    Hey Everyone, This is a little project of mine. I'm including a daigram of it - top view -> down. This is part of a larger system. Consider a fence gate made of poles. In my diagram Point "A" in blue is a non-moving vertical pole. Point "B" red is a pivot point for an arm containing all...
  3. B

    MHB What is the Angle Between Vectors Using the Dot Product Formula?

    Find the angle between the vectors v=-5\sqrt{3}i+5j and w=5i
  4. K

    Calculating Angle Between E-Field and Current Vectors in Anisotropic Mat.

    In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector. i) Calculate the angle between the...
  5. P

    Finding the angle between vectors a and m, knowing the magnitude of m and n

    Summary: Finding angle between vectors a and m, knowing magnitude of m and n, also the angle between m and n with 60 degrees. Using geometry, it looks like the angle is 30 degrees but the answer is suppose to be 54.7 degrees. I'm not sure how to solve this.
  6. opus

    I Looking for deductions to find the angle between vectors

    I have a question in my text (Intro Mechanics by Kleppner) and a question is to find the cosine and sine between two vectors. It gives me the cosine piece: $$\cos(\vec A, \vec B) = \frac{\vec A ⋅\vec B}{|A||B|}$$ which I assume is just from the dot product, but it has no derivation of this, and...
  7. Eucliwood

    B Measuring Angles: Negatives & Quadrants

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  8. S

    Find angle between displacement and velocity vector

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  9. karush

    MHB T12.3.11 Find the angle between vectors

    ok don't know the book answer but think this is ok suggestions welcome:cool: $\tiny{t12.3.11}\\$ $\textsf{Find the angle between vectors}$ \begin{align*}\displaystyle u&=\sqrt{3i}-7j\\ v&=\sqrt{3i}+j-2k\\ u \cdot v&=(\sqrt{3})(\sqrt{3}) + (-7)(1)+(0)(-2)\\ &=3-7+0\\ &=-4 \end{align*}...
  10. andylatham82

    I How to find angle between vectors from dot and cross product

    Hi, hopefully a quick question here...how do you calculate the angle between two vectors if the only information you have is the value of their scalar product and the magnitude of their cross product? Thanks! Andy
  11. andylatham82

    I Angle between vectors via scalar product vs vector product

    Hello, I have a question about why I can't determine the angle between two vectors using their cross product. Say there are two vectors in the XY-plane that we want to find the angle between: A = -2.00i + 6.00j B = 2.00i - 3.00j The method to do this would be to work out the scalar product of...
  12. karush

    MHB angle between vectors

    $\tiny{}$ $\textsf{Given $v=-7i-j$ and $w=-i-7j$}\\$ $\textsf{find the angle between v and w}\\$ $\displaystyle \frac{\left(-7, -1, 0\right)\cdot\left(-1, -7, 0\right)} {\sqrt{50}\cdot \sqrt{50}} =\frac{14}{50}\approx 0.28$ $\arccos(0.28)\approx 73.74^o$ $\textit{not sure if this is...
  13. ThomasMagnus

    Confusion over angle between vectors

    Homework Statement I have been doing dot and cross product recently. I get how to calculate everything; however, I am confused about which angle to use when asked to find the angle between two vectors. When you use the cross product, you always end up with 2 answers, for example 120° and...
  14. D

    Calculating Angle Between Vectors G & F

    I have some question guys i have four points in the x,y plane in cartesian coordinates. A (Ax,Ay) B(Bx,By) C(Cx,Cy) D(Dx,Dy) A and B is vector G C and D is vector F I would like to know what is the equation to get the angle between those two vectors (G,F) . and what are the...
  15. L

    Solving for the Angle Between Vectors A and B

    [b]1. Homework Statement Vectors A and B have equal magnitudes of 15.0. If the sum of A and B is the vector 4.55 j, determine the angle between A and B. [b]3. The Attempt at a Solution I thought I know what I was doing but apparently not. Please help!
  16. J

    Finding the Angle Between Vectors a and b

    [b]1. Homework Statement The vector a=2 and vector b=1. The vectors a+5b and 2a-3b are perpendicular. Determine the angle between a and b . Homework Equations The dot product a•b=lallblcosθ The Attempt at a Solution I've tried a few things but none of it really makes sense...
  17. M

    Angle Between Vectors A & B: Solve the Mystery!

    Homework Statement Vectors A and B have scalar product -6.00 and their vector product has magnitude 4.00. What is the angle between these two vectors?Homework Equations A \dot B = ||A|| ||B|| cos θ and A \cross B = ||A|| ||B|| sin θThe Attempt at a Solution Then I reasoned that tan(θ) = -4/6 so...
  18. O

    Angle Between Vectors given Ratio of Resultants, why is this wrong?

    Homework Statement Vectors A and B have the same magnitude. Given that the magnitude of A + B is 75 times greater than the magnitude of A - B, find the angle between them? Homework Equations We know that A=B, so: 2AB+2ABCos\theta=75(2AB-2ABCos\theta) Given that A=B...
  19. T

    Verifying Angle Between Vectors: \vec u & \vec v

    Homework Statement Using these 2 vectors: \vec u = (3,-4,0) \vec v = (1,1,1) I must verify that theta is the same with these 2 equations: Dot product \vec u \bullet \vec v = ||\vec u|| ||\vec v|| cos( \theta) Cross product ||\vec u \wedge \vec v|| = ||\vec u|| ||\vec v|| sin(...
  20. U

    Oriented angle between vectors

    "oriented" angle between vectors Say, we are given two vectors v1(1,2) and v2(2,1). Question - how to find an "oriented" angle in clockwise direction between the given vectors? Note, the angle between v1 and v2 will be equal to (360-(the angle between v2 and v1)) Any suggestions will be...
  21. S

    Basic Lin Alg - Angle Between Vectors

    Homework Statement Find the angle between the vectors [-1, 1, 2] and [2, 1, -1]. Homework Equations cos(theta) = <u, v> / ||u||||v|| The Attempt at a Solution I get <u, v> = -3 and both norms equal to the square root of six. Then cos(theta) equals (-1/12), but I don't think this...
  22. S

    If θ is the angle between vectors

    Okay, here's a cool question I'm just not able to get: If θ is the angle between vectors A = (1,1,...,1) & B = (1,2,...,n) then find the limiting value of θ when n → ∞, where n is the dimension of the space. Okay, I am using A • B = ||A|| ||B|| cos θ. Since A = (1,1,...,1) we have...
  23. H

    Find Vectors to Make 60° Angle w/ v=<3,4>

    how to find two vectors that make an angle of 60O with v=<3,4>?? can i let b=<x,y>, then cos60=(3x+4y)/5sqrt(x2+y2) 5sqrt(x2+y2)=3x+4y
  24. R

    Finding the Angle Between Vectors A & B

    Homework Statement Vectors A and B have equal magnitudes of 5.12. If the sum of A and B is the vector 6.37j, determine the angle between A and B. Homework Equations Cosine law? The Attempt at a Solution I attempted to do this using the cosine law which I thought made sense but...
  25. R

    What is the Angle Between Vectors A and C?

    Homework Statement If A = 1i + 2j + 3k and B = 1i + 2k, and if C = A X B, then the angle between the vector A and the vector C is: Homework Equations AxB = ((a2b3-a3b2)i + (a3b1-a1b3)j + (a1b2-a2b3)k) A·B = ABcosθ = AiBi + AjBj + AkBk The Attempt at a Solution I got C = 4i + j -2k...
  26. P

    How Is the Angle Calculated Between Displacement Vectors in Navigation?

    A person walks in the following pattern: 6.6 km north, then 3.0 km west, and finally 7.0 km south. How far and in what direction would a bird fly in a straight line from the same starting point to the same final point? I found that the distance would be 3.02 km. My problem is trying to find...
  27. C

    Find the angle between vectors r(t) and r'(t) Question?

    Hi everyone, i have a midterm exam tomorrow night (Thursday night) and I'm looking at problems in the section of the book to help myself prepare. Can anyone answer this question for me? I have attempted to answer this problem but without the answer i don't know if I'm right. 1. Find the angle...
  28. D

    What is the signed angle between two 2-dimensional vectors?

    The following discussion is in 2 dimensions: Take two vectors, A and B. Generally we find the angle (theta) between them by cos(theta) = dot(A,B) / (norm(A) * norm(B)) however, take vector A to be [0, 1] (straight up) If vector B is [1,1], the angle between them is pi/4 radians. If...
  29. S

    Possible What is the angle between these two vectors?

    [SOLVED] Angle between vectors: Homework Statement What is the angle between A and B ? Answer in units of degree. Homework Equations From the previous problem: A = 1.02 units long in positive y direction [0i + 1.02j] B = [-6.18i + 3.64j] The Attempt at a Solution I got the...
  30. I

    How to calculate Angle between vectors

    Hello, In attached image the three black lines represent three force vectors x,y,z of some arbitrary magnitude, these I can normalize and by taking the sin-1 I get there effect angle. What I would like to calculate is the angle between any two of the black line axies thus theta xy = ?. also I...
  31. R

    How Do You Calculate the Angle Between Two Vectors in a Sailboat Rigging System?

    Homework Statement The figure shows a mast and parts of the equipments of a sailboat. The elements CD and EF belongs to the same plane, CD has 7.5m of length and has an angle of 45º with a vertical line that passes by C. When \theta = 15º the tension in the rope AB is 230N...
  32. D

    Find angle between vectors with cosine law

    Hi I would really appreciate it if anybody could lead me in the right direction on this one... |A| = |B| |A+B| = 100|A-B| I need to find the angle between A and -B for the statement to be true. Using cosine laws I've come up with the following eqn: |A+B| = 100|A-B| 2|A|^2 + (2|A|^2)(cosx)...
  33. I

    Angle between vectors that are both off set by a third vector

    Hello, was wondering if someone could help me with a little vector maths problem please. What I need to find is the angle between two vectors that are derived from/relative to a third vector. So what I having working/understood is that theta = cos-1(“vector a” * “vector b”) where “vector...
  34. T

    What is the Angle Between Vectors Method?

    Homework Statement A little left of field this question.. http://img455.imageshack.us/img455/4531/bantuzfy5.jpg The Attempt at a Solution I'm unsure more with the wording of this question if anything rather than the method of how to go about it. How would I go about finding the...
  35. N

    Calculating the Angle Between Two Vectors

    Calculate the angle between the vectors: A = 6.8i + 4.5j + 6.2k B = 8.2i + 2.3j – 7.0k A*B= AB cos A*B=AxBx + AyBy + AzBz A*B= (6.8)(8.2) + (4.5)(2.3) + (6.2)(-7.0)=22.71 22.71 cos = 87.48 degrees Correct?
  36. B

    Finding Angle Between Vectors: Urgent Problem

    Vector Problem URGENT Two vectors A and B have precisly equal magnitudes. In order for the magnitude of A +B to be larger then the magnitude of A - B by the factor n, what must be the anle between them? There is the question i need help on this quickly thank you for nehelp i know u have to...
  37. P

    Finding the smallest angle between Vectors

    Vectors Angles How do you find the smallest angle between vectors? This one is tricky because 1 vector and 3 values and the other has 2. A (2i+j+3k) and B (-2j+2k) Thanks, Pamela