calar Triple Product
If a, b, c are three vectors, then (a * b) * c is called scalar triple product and is denoted by [a b c].
∴ [a b c] = (a * b) * c
Geometrical Interpretation of Scalar Triple Product
The scalar triple product (a * b) * c represents the volume of a parallelepiped whose coterminous edges are represented by a, b and c which form a right handed system of vectors.
Expression of the scalar triple product (a * b) * c in terms of components
a = a1i + a1j + a1k, b = a2i + a2j + a2k, c = a3i + a3j + a3k is
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Properties of Scalar Triple Products
1. The scalar triple product is independent of the positions of dot and cross i.e., (a * b) * c = a * (b * c).
2. The scalar triple product of three vectors is unaltered so long as the cyclic order of the vectors remains unchanged.
i.e., (a * b) * c = (b * c) * a= (c * a) * b
or
[a b c] = [b c a] = [c a b].
3. The scalar triple product changes in sign but not in magnitude, when the cyclic order is changed.
i.e., [a b c] = – [a c b] etc.
4. The scalar triple product vanishes, if any two of its vectors are equal.
i.e., [a a b] = 0, [a b a] = 0 and [b a a] = 0.
5. The scalar triple product vanishes, if any two of its vectors are parallel or collinear.
6. For any scalar x, [x a b c] = x [a b c]. Also, [x a yb zc] = xyz [a b c].
7. For any vectors a, b, c, d, [a + b c d] = [a c d] + [b c d]
8. [i j k] = 1
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11. Three non-zero vectors a, b and c are coplanar, if and only if [a b c] = 0.
12. Four points A, B, C, D with position vectors a, b, c, d respectively are coplanar, if and only if [AB AC AD] = 0.
i.e., if and only if [b — a c— a d— a] = 0.
13. Volume of parallelepiped with three coterminous edges a, b,c is | [a b c] |.
14. Volume of prism on a triangular base with three coterminous edges a, b,c is 1 / 2 | [a b c] |.
15. Volume of a tetrahedron with three coterminous edges a, b,c is 1 / 6 | [a b c] |.
16. If a, b, c and d are position vectors of vertices of a tetrahedron, then
Volume = 1 / 6 [b — a c — a d — a].
Gaurav Seth 4 years, 2 months ago
calar Triple Product
If a, b, c are three vectors, then (a * b) * c is called scalar triple product and is denoted by [a b c].
∴ [a b c] = (a * b) * c
Geometrical Interpretation of Scalar Triple Product
The scalar triple product (a * b) * c represents the volume of a parallelepiped whose coterminous edges are represented by a, b and c which form a right handed system of vectors.
Expression of the scalar triple product (a * b) * c in terms of components
a = a1i + a1j + a1k, b = a2i + a2j + a2k, c = a3i + a3j + a3k is
<figure>Properties of Scalar Triple Products
1. The scalar triple product is independent of the positions of dot and cross i.e., (a * b) * c = a * (b * c).
2. The scalar triple product of three vectors is unaltered so long as the cyclic order of the vectors remains unchanged.
i.e., (a * b) * c = (b * c) * a= (c * a) * b
or
[a b c] = [b c a] = [c a b].
3. The scalar triple product changes in sign but not in magnitude, when the cyclic order is changed.
i.e., [a b c] = – [a c b] etc.
4. The scalar triple product vanishes, if any two of its vectors are equal.
i.e., [a a b] = 0, [a b a] = 0 and [b a a] = 0.
5. The scalar triple product vanishes, if any two of its vectors are parallel or collinear.
6. For any scalar x, [x a b c] = x [a b c]. Also, [x a yb zc] = xyz [a b c].
7. For any vectors a, b, c, d, [a + b c d] = [a c d] + [b c d]
8. [i j k] = 1
<figure>11. Three non-zero vectors a, b and c are coplanar, if and only if [a b c] = 0.
12. Four points A, B, C, D with position vectors a, b, c, d respectively are coplanar, if and only if [AB AC AD] = 0.
i.e., if and only if [b — a c— a d— a] = 0.
13. Volume of parallelepiped with three coterminous edges a, b,c is | [a b c] |.
14. Volume of prism on a triangular base with three coterminous edges a, b,c is 1 / 2 | [a b c] |.
15. Volume of a tetrahedron with three coterminous edges a, b,c is 1 / 6 | [a b c] |.
16. If a, b, c and d are position vectors of vertices of a tetrahedron, then
Volume = 1 / 6 [b — a c — a d — a].
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