In mathematics, the product of three vectors refers to the scalar triple product of vectors. Scalar quantities are derived from this formula and are expressed as (a x b).c. The dot and cross in this formula can be interchanged, that is, (a x b).c = a.(b x c). The purpose of this article is to teach students about the definition, formula, properties and more of the scalar triple product and vector triple product.
A Proof of Scalar Triple Products
Using a scalar triple product formula, we combine the cross product of two of the vectors and the dot product of one of the vectors. We can write it as follows:
abc= (a x b).c
This formula indicates the volume of a parallelepiped with three coterminous edges, for example, a, b, and c. In terms of the volume, the cross product of two vectors (let a and b be the vectors) results in the volume of the base. As a result, we get a perpendicular direction to both vectors. By calculating the height along the direction of the resultant cross product we can find the third vector (say c).
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In other words, the parallelogram’s area is a product of |a x b|, and the direction the vector faces is perpendicular to the base.
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The height is denoted by |c| cos cos Ф, where Ф denotes the angle between a x b and c.
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The diagram above shows that the direction of the vector |a x b| is perpendicular to the base of the diagram, and denotes the height as |c| cos cos Ф.
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By defining the expansion of vector cross products, calculating the scalar triple product proof becomes a breeze.
Let a= a[_{1}][hat{i}] + a[_{2}][hat{j}] + a[_{3}][hat{k}], b= b[_{1}][hat{i}] + b[_{2}][hat{j}] + b[_{3}][hat{k}], c = c[_{1}][hat{i}] + c[_{2}][hat{j}] + c[_{3}][hat{k}]
Now, (a x b) . c = [begin{vmatrix} hat{i} & hat{j} & hat{k}\ a_{1} & a_{2} & a_{3}\
b_{1} & b_{2} & b_{3} end{vmatrix}]. (c[_{1}] [hat{i}] + c[_{2}][hat{j}] + c[_{3}][hat{k}])
(a x b). c = [begin{vmatrix}hat{i}.(c_{1} hat{i} + c_{2}hat{j} + c_{3}hat{k}) & hat{j}. (c_{1} hat{i} + c_{2}hat{j} + c_{3}hat{k}) & hat{k}.(c_{1} hat{i} + c_{2}hat{j} + c_{3}hat{k})\ a_{1} & a_{2} & a_{3}\ b_{1} & b_{2} & b_{3} end{vmatrix}]
From the properties of the dot product of vectors:
[hat{i}]. [hat{i}] = [hat{j}]. [hat{j}] = [hat{k}]. [hat{k}] = 1 (cos 0 = 1)
It implies [hat{i}]. (c[_{1}] [hat{i}] + c[_{2}][hat{j}] + c[_{3}][hat{k}]) = c[_{1}]
[hat{j}]. (c[_{1}] [hat{i}] + c[_{2}][hat{j}] + c[_{3}][hat{k}]) = c[_{2}]
[hat{k}]. (c[_{1}] [hat{i}] + c[_{2} hat{j}] + c[_{3}][hat{k}]) = c[_{3}]
(a x b) . c = [begin{vmatrix} c_{1} & c_{2} & c_{3}\ a_{1} & a_{2} & a_{3}\ b_{1} & b_{2} & b_{3} end{vmatrix}]
(a x b) . c = [begin{vmatrix} c_{1} & c_{2} & c_{3}\ a_{1} & a_{2} & a_{3}\ b_{1} & b_{2} & b_{3} end{vmatrix}]
Scalar Triple Product Properties
The scalar triple product is cyclic; that is;
abc = bca = cab = -bac = -cba = -acb
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If the vectors taken in scalar triple product definition, say a, b, and c are cyclically permuted, then:
(a x b).c = a.(b x c)
For any k that belongs to Real number,
Ka kb kc = kabc
= a.(cxd)+b.(cxd)
= acd + bcd
Analysing the scalar triple product formula, some conclusions can be drawn:
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Scalar triple products always produce scalar quantities as their resultant.
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Scalar triple product formulas are determined by calculating cross products of two vectors. Thereafter, the dot product of the remaining vector and the resultant vector is calculated.
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This suggests one of the three vectors taken is equal to zero magnitudes if the triple product is zero.
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A parallelepiped can be easily calculated by using this method.