त्रिक गुणनफल

सदिश बीजगणित में तीन ३-विमीय सदिशों का गुननफल त्रिक गुणनफल (triple product) कहलाता है। दो तरह के त्रिक गुणनफल होते हैं- अदिश मान वाला त्रिक गुणनफल तथा सदिश त्रिक गुणनफल।

अदिश त्रिक गुणनफल

${a}\cdot ({b}\times {c})={c}\cdot ({a}\times {b})={b}\cdot ({c}\times {a})$

इस गुणन्फल का मान तीनों सदिशों $\mathbf {a}$, $\mathbf {b}$ और $\mathbf {c}$ से बने हुए समान्तरषटफलक के आयतन के बराबर होता है।

गुण

• चक्रीय शिफ्ट करने पर अदिश त्रिक गुणनफल (a, b, c) का मान नहीं बदलता। :

  $\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )$

• Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product.

  $\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv (\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c}$

• Swapping any two of the three operands negates the triple product. This follows from the circular-shift property and the anticommutativity of the cross product.

  ${\begin{aligned}&\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\\\equiv -&\mathbf {a} \cdot (\mathbf {c} \times \mathbf {b} )\\\equiv -&\mathbf {b} \cdot (\mathbf {a} \times \mathbf {c} )\\\equiv -&\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )\end{aligned}}$

• The scalar triple product can also be understood as the determinant of the 3×3 matrix (thus also its inverse) having the three vectors either as its rows or its columns (a matrix has the same determinant as its transpose):

  $\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv \det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}={\rm {det}}\left(\mathbf {a} ,\mathbf {b} ,\mathbf {c} \right).$

• If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume.
• If any two vectors in the triple scalar product are equal, then its value is zero:

  $\mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )\equiv \mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )\equiv \mathbf {a} \cdot (\mathbf {b} \times \mathbf {b} )\equiv \mathbf {a} \cdot (\mathbf {a} \times \mathbf {a} )\equiv 0$

• Moreover,

  $[\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )]\mathbf {a} \equiv (\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )$

• The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products:

  $([\mathbf {a} \times \mathbf {b} ]\cdot \mathbf {c} )\;([\mathbf {d} \times \mathbf {e} ]\cdot \mathbf {f} )\equiv \det \left[{\begin{pmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{pmatrix}}\cdot {\begin{pmatrix}\mathbf {d} &\mathbf {e} &\mathbf {f} \end{pmatrix}}\right]\equiv \det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {d} &\mathbf {a} \cdot \mathbf {e} &\mathbf {a} \cdot \mathbf {f} \\\mathbf {b} \cdot \mathbf {d} &\mathbf {b} \cdot \mathbf {e} &\mathbf {b} \cdot \mathbf {f} \\\mathbf {c} \cdot \mathbf {d} &\mathbf {c} \cdot \mathbf {e} &\mathbf {c} \cdot \mathbf {f} \end{bmatrix}}$

  This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product.

सदिश त्रिक गुणनफल

यह गुणनफल एक सदिश राशि होती है।

a*(b*c)=(a.c)b-(a.b)c