What is Vector cross product: Definition and 39 Discussions

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in three-dimensional space



{\displaystyle \mathbb {R} ^{3}}
, and is denoted by the symbol


{\displaystyle \times }
. Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
If two vectors have the same direction or have the exact opposite direction from one another (i.e., they are not linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths.
The cross product is anticommutative (i.e., a × b = − b × a) and is distributive over addition (i.e., a × (b + c) = a × b + a × c). The space



{\displaystyle \mathbb {R} ^{3}}
together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to a pseudovector, or the exterior product of vectors can be used in arbitrary dimensions with a bivector or 2-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in n dimensions, take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. (See § Generalizations, below, for other dimensions.)

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  1. A

    Condition for coplanarity of two lines

    So I tried to solve it this way: The equations of the lines in vector form are $$(x-2)\hat i+(y-3)\hat j+(z-4)\hat k=\lambda (\hat i+\hat j-K\hat k)$$ $$(x-1)\hat i+(y-4)\hat j+(z-5)\hat k=\mu (K\hat i+2\hat j+1\hat k)$$ Since the lines are some real multiple of the vectors, For coplanarity...
  2. nafisanazlee

    What Is the Correct Angle Between Vectors a and c?

    The solution to the question is attached herewith. I approached in the exact same way and got |c| = 2. Then I thought like this: the angle between a and a×b is 90°, and the angle between c and a×b is 30° (given). So one of the possibilities is, the angle between a and c is 90-30=60° degree. |a|...
  3. L

    What is the formula for the norm of a vector cross product?

    Hi everyone, I'm having problems with task c In the task, the norm has already been defined, i.e. ##||\vec{c}||=\sqrt{\langle \vec{c}, \vec{c} \rangle }## I therefore first wanted to calculate the scalar product of the cross product, i.e. ##\langle \vec{a} \times \vec{b} , \vec{a} \times...
  4. D

    I Question about the vector cross product in spherical or cylindrical coordinates

    Hi If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b. Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...
  5. greg_rack

    Vector cross product anti-commutative property

    That may sound really silly, and that may be due to my lack of understanding of the operations itself, but: if ##|\vec{a}\times\vec{b}|=|\vec{a}|\cdot|\vec{b}|sin\theta##, being ##\theta## the angle between the two vectors, how could ##\vec{b}\times\vec{a}## be different? Wouldn't it be just the...
  6. greg_rack

    Calculating vector cross product through unit vectors

    Writing both ##\vec{U}## and ##\vec{B}## with magnitude in all the three spatial coordinates: $$ \vec{U}\times \vec{B}= (U_{x}\cdot \widehat{i}+U_{y}\cdot \widehat{j}+U_{z}\cdot \widehat{k})\times (B_{x}\cdot \widehat{i}+B_{y}\cdot \widehat{j}+B_{z}\cdot \widehat{k})$$ From this point on, I...
  7. J

    Vector Cross Product With Its Curl

    Starting with LHS: êi εijk Aj (∇xA)k êi εijk εlmk Aj (d/dxl) Am (δil δjm - δim δjl) Aj (d/dxl) Am êi δil δjm Aj (d/dxl) Am êi - δim δjl Aj (d/dxl) Am êi Aj (d/dxi) Aj êi - Aj (d/dxj) Ai êi At this point, the LHS should equal the RHS in the problem statement, but I have no clue where...
  8. H

    I What is the purpose of cross multiplication in vector multiplication?

    Hi, what does it mean to cross multiply two vectors? I couldn't imagine them in real life. eg Force vector. Multiplying Force vector to a scalar value means you multiple the 'Strength' of the force, Dot multiplication of Force with displacement to get work, means you get the work in...
  9. B

    What is the Direction of A X B Using the Right Hand Rule?

    Homework Statement The direction of vectors A and B are given below for several cases. For each case, state the direction of A X B. a) A points east, B points south. b) A points east, B points straight down. c) A points straight up, B points north. d) A points straight up, B points straight...
  10. Muthumanimaran

    Vector Cross Product Homework: Expand $\vec{v}\times({\nabla}{\times}\vec{A})$

    Homework Statement This is not a homework problem, I am currently reading the Derivation of potential of a charged particle in Electric and Magnetic field from the book Mechanics by Symon (I attached the image of the page), I need to know how to expand the vector cross product such as...
  11. parshyaa

    I Vector Cross Product: Understanding the Perpendicular Result

    Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.
  12. H

    Vector cross product with curl

    Homework Statement Using index-comma notation only, show: \begin{equation*} \underline{\bf{v}} \times \text{curl } \underline{\bf{v}}= \frac{1}{2} \text{ grad}(\underline{\bf{v}} \cdot \underline{\bf{v}}) - (\text{grad } \underline{\bf{v}}) \underline{\bf{v}} \end{equation*} Homework Equations...
  13. P

    MHB Did my book do this wrong? (Vector Cross Product)

    Reading a book about 3d math, and I am confused as to what happened on this Vector Cross Product problem. I'm thinking there was just an error that wasn't caught. For the first row, instead of (3)(8)-(-4)(-5) shouldn't it have been (3)(8)-(4)(-5) and had the same displayed result of 44? And for...
  14. O

    Cross product of 2 vectors of same magnitude

    Homework Statement Vectors A and B both have magnitude M. Joined at the tails, they create a 30' angle. What is A x B in terms of M? Homework EquationsThe Attempt at a Solution 0? OR M^2? Sqrt(3)M/3?
  15. C

    Vector Cross Product Homework: Find a×(a-2b+c)

    Homework Statement Given a×b=-i-j+3k and c×a=2i-3j+k, find a×(a-2b+c) Homework Equations Cross product (DONE WITHOUT MATRICES). The Attempt at a Solution a[/B]×b=c=-(b×a)is all I'm getting to at this point
  16. N

    Levi Civita symbol on Curl of Vector cross product

    Homework Statement Use the LC symbol to calculate the following: $$\nabla \times \frac{\vec{m} \times \hat{r}}{r^2}$$ Where ##\vec{m}## is just a vector, and ##\hat{r}## is the unit radial vector and ##r## is the length of the radial vector. Homework Equations On the Levi Civita symbol...
  17. F

    Finding the Angle Between Vectors A and B in the Cross Product

    Homework Statement Vectors A & B lie in an xy plane. A has a magnitude 7.4 and an angle 142(deg) with respect to the +x direction. B has components (-6.84i, -7.37j) B) What is the angle between the -y axis and the direction of the Cross product between A and B? Homework Equations Cross...
  18. B

    A Nice Vector Cross Product Proof.

    Homework Statement If a, b, c, d are all vectors contained in the same plane, explain why (a X b) X (c X d) = <0,0,0>Homework Equations The Cross Product! The Attempt at a Solution I know that since all of the vectors are in the same plane that means that a cross product between any of the...
  19. J

    Proving Vector Cross Product Properties in ℝ3?

    If e1 and e2 are vectors in ℝ3 show that e1 x e2 = e3, e2 x e3 = e1 and e3 x e1 = e2. I have tried to prove this but I can't get it. My attempt: Step 1: [a1, a2, a3] x [b1, b2, b3] = [a2b3-a3b2, a3b1-a1b3, a1b2-a2b1] Step 2: [b1, b2, b3] x [a2b3-b2a3, a3b1-a1b3, a1b2-a2b1] =...
  20. F

    Finding the Magnitude of Cross Product Vector Question

    Homework Statement From John Taylor's Classical Mechanics: Show that definition (1.9) of the cross product is equivalent to the elementary deinition that R x S is perpendicular to both R and S, with magnitude rssinθ and direction given by the right hand rule. [Hint: It is a fact (though...
  21. G

    Torque- Vector cross product using both geometric and algebraic methods

    Homework Statement A lever is orientated along the y direction in a Cartesian coordinate system. The length of the lever is 0.5m and one end of it is at the origin of the coordinate system. A (3i-5j)N force applied to the other end of the lever. Calculate the Torque produced by the force...
  22. majormaaz

    Physics Vector Cross Product problem

    1. Homework Statement Two vectors are given by A = -6 i + 5 j and B = 1 i + 4 j Find A X B (answer only in terms of i, j, k) Find the angle between A and B (answer is terms of degrees) 2. Homework Equations All I was told was that if I set a 3x3 matrix like this: i j k -6 5 0 1 4 0...
  23. S

    Vector cross product with coefficients

    Anyone know how would I simplify a cross product where the two vectors have coefficients? For example (x/(y^3))\bar{r} X (x/(y))\bar{L} Thanks!
  24. H

    Solving for R2: Calculating Vector Cross Product

    I am trying to figure out the following If R2 = 1.043j -1.143k Then how can R2 = 1.547
  25. T

    Aligning 3 point plate - vector cross product useful?

    Homework Statement Hello, I have a plate that needs levelling. I can only twist the back 2 feet threaded feet, while the single front foot is fixed, as in the diagram: As mentioned above I can only TWIST the back two feet (separately or together) causing the plate to tilt around the...
  26. S

    Origin of Vector Cross Product

    I understand the cross product of vectors to some degree and i can calculate. But i don't really understand the origin of the cross product What does a vector cross product mean in physical terms. Vector addition is quite easy to understand. I don't think the cross product is 'multiplication...
  27. X

    Vector Cross Product Homework: Find 3rd Vector Perpendicular to C & D

    Homework Statement C= B|A| + A|B| D= A|B|-B|A| C and D are orthogonal Find a third vector perpendicular to both C and D Homework Equations [AxB] = |A||B|sin(theta) The Attempt at a Solution I know that to find the answer I need to find the cross product of C and D. I have done...
  28. N

    Vector cross product identity proof

    Homework Statement \bigtriangledown\times\\(v\times w)= v(\bigtriangledown\cdot w) - w(\bigtriangledown\cdot v)+ (v\cdot\bigtriangledown)w - (w\cdot\bigtriangledown) v I've tried expanding left side and get [v1(dw2/dy+dw3/dz)-w1(dv2/dy+dv3/dz)]i +...
  29. D

    Vector Cross Product (with Right Triangle)

    Homework Statement For the vectors in the figure below, with a = 8, b = 7, c = sqrt(113), give the magnitude and direction of the following cross products. (See attachment for figure/right triangle). (a) a x b (b) a x c (c) b x c Homework Equations \vec{A} \cdot...
  30. R

    Vector Cross Product: Calculating -i x i = 0?

    What is the cross product of -i x i? Is it negative 1 or is still just 0?
  31. G

    How can the vector cross product be calculated when only magnitudes are given?

    [b]1. Find axb: Given the magnitude of lal=3 and the magnitude of lbl=2. [b]2. Since I have the magnitudes, I thought maybe I could use the equation of axb=lal lbl sin theta. [b]3. I thought since I am trying to find axb that I could use 90 as the angle theta to find axb. I am...
  32. H

    Vector Cross Product: Perpendicular Vectors lV_1 x V_2l

    If vectors V_1 and V_2 are perpendicular, lV_1 x V_2l =? I know that if they are parallel for vector cross product, they equal 0.
  33. A

    Vector Cross Product Formula in Spherical & Cylindrical Coordinates

    Hello all, it might be funny! but i am stuck to it! what is the vector cross product formula in spherical and cylindrical coordinates?! I know for Cartesian coordinate we have that nice looking determinant. but what about the other coordinates. I had looks to all the math books (like...
  34. J

    Vector Cross Product: R4 Vectors & Permutation Symbol

    Is there such a thing as a cross product for R4 vectors? Can you use the permutation symbol to express it in the same way that it can be expressed in R3? Would the correct way to write it be: e _{i,j,k,l} u _{j} v _{k}?
  35. N

    Vector Cross Product Homework: Solving AxB with B values of 8i+16j and -8i-16j

    Homework Statement Given that A = 2i + 4j, evaluate each of the following. (Hint: This question can be answered without computation.) (a) What is AxB when B = 8i + 16j? (b) What is AxB when B = -8i - 16j? Homework Equations AxB=(Axi + Ayj) x (Bxi +Byj) =(AxBx)(i x i)...
  36. V

    Finding torque (vector cross product)

    A particle position is described by position vector r = 3i + 2j and the force vector i - 2j acts on the object. 1) Find the torque about an axis through the origin and perpendicular to the xy plane. Draw the two vectors to check your torque direction. I used the right hand rule and found...
  37. P

    Understanding Vector Cross Product: Finding the Angle Between Two Vectors

    This is the question: Two vectors A and B have magnitude A = 3 and B = 3. Their vector product is A X B = -5k+2i. What is the angle Between A and B. OK so I'll start with what I do know. I do know that the cross product is the magnitude of A times magnitude of B times sin theta of B. I end...
  38. E

    Rules regarding the vector cross product and dot product

    hi, I'm currently doing a mechanics module at Uni. The thing is, I'm not very sure about rules regarding the vector cross product and dot product. For example, it says in my notes for angular momentum: "Introducing polar coordinates \mathbf{r} = r(cos \Phi \mathbf{i} + sin \Phi...
  39. V

    Right hand rule in vector cross product

    I have learned just about the right hand rule in vector cross product. How is this proved? Can anybody give an example where the cross product plays an important role and where the vector cross product formula is obeyed? Our professor just told us that the torque due to a force acting on a body...