What is Vector triple product: Definition and 16 Discussions

In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.

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  1. A

    Given value of vectors a,b, b.c and a+(b×c), Find (c.a)

    I thought this was too easy $$a+(b\times c)=0\implies a=-(b\times c)=(c\times b)$$ Then $$3(c.a)=3(c.(c\times b))=0$$ Since cross product of vectors is perpendicular to both vectors and dot product of perpendicular vectors is zero. Now here's the problem, correct answer given is 10. But how do...
  2. Decimal

    I Understanding the Vector Triple Product Proof

    Hello, I am having trouble understanding a proof presented here: http://www.fen.bilkent.edu.tr/~ercelebi/Ax(BxC).pdf This is a proof of the triple product identity, but I don't understand the last step, where they calculate ##\lambda##. Don't you lose all generality when you state ##\vec A##...
  3. UAJalen

    I Can you reduce a vector triple product? i.e. (A x (uB x C))

    My question is simply whether you can reduce a vector triple product, or more generally a scalar multiplier of a vector in a cross product? Given: (A x (uB x C) = v, where u and v are known constants. Is it valid to change that to: u(A x (B x C) = v or (A x uB) = v, can you change that to u(A...
  4. R

    I Vector Triple Product - Physcial Significance

    Hii, As we know, Scaler triple product is volume of parallelopiped constructed by its three sides. Similary, What is the physical significance and geometrical interpretation of Vector triple product ? Also, What are the application where we use such mathematics and why ? Regards, Rahul
  5. M

    A Exterior Algebra Dual for Cross Product & Rank 2 Tensor Det

    The determinant of some rank 2 tensor can be expressed via the exterior product. $$T = \sum \mathbf{v}_i \otimes \mathbf{e}_i \;\;\; \text{or}\sum \mathbf{v}_i \otimes \mathbf{e}^T_i $$ $$ \mathbf{v}_1\wedge \dots \wedge \mathbf{v}_N = det(T) \;\mathbf{e}_1\wedge \dots \wedge\mathbf{e}_N$$ The...
  6. BobJimbo

    Resolving Vectors Using the Vector Triple Product

    The problem: By considering w x (p x w) resolve vector p into a component parallel to a given vector w and a component perpendicular to a given vector w. Hint: a x (b x c) = b(a x c) - c(a x b) I'm afraid I really have no idea where to go with this one. The hint leads to: p(w.w) - w(w.p) =...
  7. D

    I Vector Triple Product: Are a & c Parallel or Collinear?

    Hi all got a confusion In many books I saw , authors used a specific statement here is it a,b,c are vectors and axb is (" a cross b") In general (axb)xc ≠ ax(bxc) but if (axb)xc = ax(bxc) solving it we get bx(axc)=0 then it implies either b is parallel to (axc) or a and c are collinear...
  8. kostoglotov

    Vector triple product causing a contradiction in this proof

    Homework Statement Prove the following identity \nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \vec{F} \times (\nabla \times \vec{G}) + \vec{G}\times (\nabla \times \vec{F}) Homework Equations vector triple product \vec{a} \times (\vec{b}...
  9. Seaborgium

    (edit:solved) Vector Triple Product, Components Parallel and Perpendicular

    Homework Statement By considering A x (B x A) resolve vector B into a component parallel to a given vector A and a component perpendicular to a given vector A. Homework Equations a x (b x c) = b (a ⋅ c) - c (a ⋅ b) The Attempt at a Solution I've applied the triple product expansion and...
  10. PcumP_Ravenclaw

    Linearity in the Vector triple product

    Dear all, I am trying to understand the vector triple product. ## x\times (y \times z) ## As the vector triple product of x,y and z lies in the plane ## (y \times z) ## the vector ## x\times (y \times z) ## can be written as a linear combination of the vectors ## \pm y ## & ## \pm z## In the...
  11. E

    What is the geometric interpretation of the vector triple product?

    The interpretation of the vector product is the area of the parallelogram with sides made up of a and b and the scalar triple product is the volume of the parallelpiped with sides a, b, and c, but what is the interpretation of the vector triple product. Is it just simply the area of the...
  12. Saitama

    How can the vector triple product be used to derive other vector products?

    I am currently going through the book Introduction Of Electrodynamics by Griffiths. I have come across vector triple product which is stated as follows in the book: $$\textbf{A} \times (\textbf{B} \times \textbf{C})=\textbf{B}(\textbf{A}\cdot \textbf{C})-\textbf{C}(\textbf{A}\cdot...
  13. H

    Geometric Proof of Vector Triple Product: Find Coefficients b and c

    Homework Statement This isn't a coursework question. Rather, I'm asking for help on a geometric proof of the vector triple product. I find it strange and annoying that I can't find this proof anywhere online, because everyone just uses the messy expansion proof, and I hate that proof because...
  14. C

    Solving the Vector Triple Product Equation: Deduction

    i) Show that: a x ( b x c) + b x ( c x a) + c x (a x b ) =0 I managed to this, by expanding each term using the definition of the triple vector product i.e. a x ( b x c) = (a.c)b-(a.b)c and adding the results. ii) and deduce that a x { b x ( c x d ) } + b x { c x ( d x a ) } + c x { d x (...
  15. H

    Vector Algebra - Vector Triple Product Proof

    Homework Statement Prove, by writing out in component form, that \left(a \times b \right) \times c \equiv \left(a \bullet c\right) b - \left(b \bullet c\right) aand deduce the result, \left(a \times b\right) \times c \neq a \times \left(b \times c\right), that the operation of forming the...
  16. R

    How to Prove the Vector Triple Product Identity?

    Homework Statement Prove that u x (v x w) = (u*w)v - (u*v)w Homework Equations I've been trying to get this one but keep ending up no where. I've tried the normal algebraic properties of the cross product but they lead me to a dead end. What I am trying right now is...