What is Scalar triple product: Definition and 15 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. Rotnort

    Volume of a Parallelepiped

    Homework Statement The volume of a parallelepiped defined by the vectors w, u, \text{ and }v, \text{ where } w=u \times v is computed using: V = w \cdot (u \times v) However, if the parallelepiped is defined by the vectors w-u, u, \text{ and }v, \text{ where } w=u \times v instead, the volume...
  2. E

    Determinant of 3x3 matrix equal to scalar triple product?

    The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product. I want to show that: ##det A \overset{!}{=} a_1 \cdot (a_2 \times...
  3. kostoglotov

    Question about Cyclical Matrices and Coplanarity of Vectors

    MIT OCW 18.06 using Intro to Linear Algebra by Strang So I was working through some stuff about Cyclic Matrices, and the text was talking about how the column vectors that make up this cyclic matrix, shown here, are coplanar, and that is the reason that Ax = b will have either infinite...
  4. kostoglotov

    Multiple Integral Challenge Question, I just need a hint

    Homework Statement I will just post an image of the problem and here's the link if the above is too small: http://i.imgur.com/JB6FEog.png?1Homework EquationsThe Attempt at a Solution I've been playing with it, but I can't figure out a good way to "grip" this problem. I can see some things...
  5. pastoreerrante

    Scalar triple product invariance under circular shift proof

    Homework Statement Prove that for any three vectors ##\hat a, \hat b ## and ## \hat c##, ##\hat a \cdot (\hat b \times \hat c)## = ##(\hat a \times \hat b) \cdot \hat c ## Homework Equations [/B] ## \hat i \cdot \hat i = \hat j \cdot \hat j = \hat k \cdot \hat k = (1)(1)\cos(0) = 1 ## ##...
  6. PcumP_Ravenclaw

    Scalar triple product and abstract vector space

    Dear all, Can anyone please explain how the linear combination of non-coplanar and non-orthogonal coordinate axes representing a point x as shown below is derived. Please use the reference text attached in this post to explain to me as i will find it a bit relevant. I want to...
  7. PcumP_Ravenclaw

    Understanding Scalar Triple Product (STP)

    Dear all, My question is from the text of Alan F. Beardon, Algebra and Geometry concerning the scalar triple product. I have attached the text in this post. In order for the STP to be non-zero. The 3 vectors must be distinct and they are not coplanar. 2 vectors can be coplanar...
  8. P

    What are a, b, and c in volume calculations for prisms and pyramids?

    The volume of a triangular prism is given by: v = ½ |a • b x c| Where b and c are two of the sides of the triangular face of the prism, and a is the length of the prism. The volume of a rectangular/parallelogram-based pyramid is given by: V = ⅓ |a • b x c| My question is, what are a, b...
  9. V

    Derivative of scalar triple product

    Homework Statement If u(t) = σ(t) . [σ'(t) x σ''(t)], show that u'(t) = σ(t) . [σ'(t) x σ'''(t)]. Homework Equations The rules for differentiating dot products and cross products, respectively, are: d/dt f(t) . g(t) = f'(t) . g(t) + f(t) . g'(t) d/dt f(t) x g(t) = f'(t) x g(t) +...
  10. S

    Scalar Triple Product Derivative

    Homework Statement Find an expression equivalent for the derivative of the scalar triple product a(t) . (b(t) x c(t))The Attempt at a Solution Initially I figured since whatever comes out of B X C is being dotted with A, I can use the derivative rules of a dot product: (a(t)' . (b(t) x...
  11. D

    A way to express scalar triple product from inter-vector angles?

    Hi, I'm trying to find a general expression for the scalar triple product for 3 vectors in a simultaneous configuration, that depends only on the inter-vector angles, A1, A2 and A3. I have expressed this quantity in terms of the spherical polar coordinates of the vectors (the length being...
  12. A

    Proving (A X B) X C = AxBxCx (i x k) + AyBxCy (j x k) | Unit Vectors

    Homework Statement Prove that (A X B) X C = AxBxCx (i x k) + AyBxCy (j x k) where i , j and k are unit vectors ? Homework Equations The Attempt at a Solution let A= Axi+Ayj B= Bxi+Byj L.H.S = (AXB)XC = (C.A) B - (C.B) A = (AxCx+AyCy)(Bxi+Byj)- (BxCx+ByCy)(Axi+Ayj)...
  13. N

    Scalar triple product coplanarity

    Homework Statement Show that u, v, w lie in the same plane in R3 if and only if u · (v × w) = 0. Homework Equations The Attempt at a Solution if u · (v × w) = 0, then u is orthogonal to vxw, and vxw is orthogonal to v and w. therefore, u must lie in the same plane...
  14. D

    Scalar triple product, volume, and ordering

    Greetings all, I'm reading about a way to solve for the volume of a "parallelepiped" in 3 space, which is determined by vectors u, v and w. The volume is apparently the absolute value of the determinant given by the matrix u1 u2 u3 v1 v2 v3 w1 w2 w3 which is the same as the scalar triple...