Scalar triple product of vectors: Derivation & Important Questions

Scalar triple product of three vectors a, b, and c refers to the dot product of a single vector with the cross product of the two remaining vectors is the. Because changing the direction of one of the vectors changes the sign of the scalar, the result is a pseudoscalar. The scalar triple product of vectors is referred to as the product of three vectors in mathematics. This formula yields scalar quantities, which are written as (a x b). c. In this formula, the dot and cross can be swapped out (a x b). c is the same as a. (b x c). It's a scalar product since it evaluates to a single number, exactly like the dot product. In this article, we will learn more about the scalar triple product of vectors.

Read More: Symmetric Matrix

Key Takeaways: Scalar Triple Product, dot product, cross product, Parallelepiped, Coterminus Edges, Vector, Coplanar, scalar quantity

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Scalar triple product of vectors

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The scalar triple product of vectors is the product of three vectors. It's a scalar product because, like the dot product, it evaluates to a single number. It entails multiplying one of the vectors' dot products by the cross product of the other two. Its mathematical expression like: (a x b). c. The volume of a parallelepiped is represented by the scalar triple product. The scalar triple product (also known as the mixed product, box product, or triple scalar product) is the dot product of one vector with the cross product of the other two vectors. 

Read More: Inverse Trigonometric Formulas

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Formula of Scalar triple product of vectors

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The product of three vectors, or the dot product of a vector with the cross product of the other two vectors, is known as the scalar triple product formula when vectors are multiplied and a scalar triple product is obtained. It's written like this:

\([\text{a b c}] =(a\times b).c\)

In this instance (a x b). c is a scalar quantity and the resultant vector. This formula can alternatively be expressed by swapping the cross and dot in the middle, as seen below (a x b). c is the same as a. (b x c). The scalar triple product of vectors is obvious from its name: it is the product of three vectors. It requires multiplying one of the vectors by the cross product of the other two.

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Geometric Derivation

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The three vectors are represented in this diagram by the coterminous edges, as illustrated. The area of the base is given by the cross product of vectors a and b, and the direction of the cross product of vectors is perpendicular to both vectors. The height in this situation is given by the component of vector c along the direction of the cross product of a and b because volume equals the product of area and height. c cos is the component's name.

Geometric Derivation of Scalar Triple Product of Vector

As a result, if the coterminous edges of a parallelepiped are designated by three vectors and a, b, and c, we can deduce that

Volume of parallelepiped = (a × b) c cos α = (a × b) . c 

as α denotes the angle between (a × b) and c. 

The extension of vector cross products is something we're known with. Bearing in mind, if it is given a = a₁\(\widehat{i}\) +a₂\(\widehat{j}\) + a₃\(\widehat{k}\), b = b₁\(\widehat{i}\) + b₂\(\widehat{j}\) + b₃\(\widehat{k}\) and c = c₁\(\widehat{i}\) +c₂\(\widehat{j}\) + c₃\(\widehat{k}\), then we can write the following equation-  

(a x b) . c =\( \begin{matrix} | \widehat{i} & \widehat{j} & \widehat{k} |\\ | a1 & a2 & a3|\\ | b1 & b2 & ib3| \end{matrix} \) . (c₁\(\widehat{i}\) +c₂ \(\widehat{j}\) + c₃\(\widehat{k}\))

The dot product of two vectors is indicated by this symbol. We can expand the previous equation using determinant properties-
(a x b) . c = \( \begin{matrix} | \widehat{i}(c1\widehat{i} +c2 \widehat{j} + c\widehat{k}) & \widehat{j}(c1\widehat{i} +c2\widehat{j} + c3\widehat{k}) & \widehat{k}(c1\widehat{i} +c2\widehat{j}+ c3\widehat{k}) |\\ | a1 & a2 & a3|\\ | b1 & b2 & ib3| \end{matrix} \)

As per the dot product of vector properties, 

\(\widehat{i }\).\(\widehat{i }\) = \(\widehat{j}\).\(\widehat{j}\) = \(\widehat{k}\).\(\widehat{k}\) = 1 as cos 0 = 1

⇒\(\widehat{i}\) (c₁\(\widehat{i}\) + c₂\(\widehat{j}\) + c₃\(\widehat{k}\)) = c₁

⇒\(\widehat{j}\) (c₁\(\widehat{i}\) + c₂\(\widehat{j}\) + c₃\(\widehat{k}\)) = c₂

⇒\(\widehat{k}\) (c₁\(\widehat{i}\) + c₂\(\widehat{j}\) + c₃\(\widehat{k}\)) = c₃

⇒(a x b) . c = \( \begin{matrix} | c1 & c2 & c3|\\ | a1 & a2 & a3|\\ | b1 & b2 & ib3| \end{matrix} \)

\(\begin{matrix} [ a & b & c ]\\ \end{matrix} \)= \( \begin{matrix} | c1 & c2 & c3|\\ | a1 & a2 & a3|\\ | b1 & b2 & ib3| \end{matrix} \)

Read More: Calculus Formula

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Properties of Scalar triple product of vectors

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The properties of scalar triple product of vectors are:

• If the vectors are permuted cyclically, then (a × b) . c = a.( b × c)
• If the vectors are permuted cyclically, then a.(b × c) = b.(c × a) = c.(a × b)
• If the triple product of vectors is zero, it can be assumed that the vectors are coplanar.

The volume of a parallelepiped is represented by the triple product. If it's zero, then such a situation could only occur if one of the three vectors has a magnitude of zero. The cross product of a and b has a direction that is perpendicular to the plane that contains a and b. Only if the vector c is also in the same plane will the dot product of the resultant with c be zero. Because the angle between the resultant and C will be 90° and cos 90°, this is the case.

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Outcomes drawn

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The outcomes drawn from scalar triple product of vectors are:

• The consequence of scalar triple products is always scalar quantities.
• Calculating the cross-products of two vectors yields scalar triple product formulas. After that, compute the dot product of the remaining vector with the resultant vector.
• If the triple product is 0, one of the three vectors taken is equivalent to zero magnitudes.
• Using this method, a parallelepiped can be simply calculated.

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Things to Remember

• The product of three vectors, or the dot product of a vector with the cross product of the other two vectors, is known as the Scalar triple product formula when vectors are multiplied and a scalar triple product is obtained.
• The mathematical expression for the scalar triple product for vectors is (a x b).c. I.n this instance (a x b). c is a scalar quantity and the resultant vector.
• The absolute value |(a×b)⋅c| of a parallelepiped's volume spanned by a, b, and c is shown by a scalar triple product (i.e., vectors a,b and c are adjacent sides a parallelepiped.)
• The volume of a parallelepiped whose three neighboring sides are the three vectors a, b, and c is calculated using the scalar triple product formula.
• The area of the base is determined by the cross-product of two vectors (let a and b) among these three. The direction of the resultant is perpendicular to both vectors.
• The height of the resultant cross product is determined by the third vector component, 'c'.
• The area of the parallelepiped is calculated by |a x b|, with the direction of the resultant vector perpendicular to the base, and the height is calculated by |c| cos \(\Phi\), where \(\Phi\) represents the angle between (a x b) and c.