Scalar Triple Product
a·(b×c) is the scalar triple product (box product). It equals the signed volume of the parallelepiped spanned by a, b, c. Zero when vectors are coplanar. Cyclic: a·(b×c) = b·(c×a) = c·(a×b).
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|a·(b×c)| = volume of parallelepiped. Coplanar vectors: a·(b×c) = 0. Cyclic permutation preserves value.
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Why: Scalar triple product gives volume of parallelepiped, tests for coplanarity (zero = coplanar), and appears in Jacobian determinant for volume change. Used in physics and differential geometry.
How: Compute b×c first, then a·(b×c). Equals determinant of matrix with a, b, c as rows. Positive = right-handed; negative = left-handed. Zero = coplanar vectors.
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Scalar Triple Product Calculator
Calculate the scalar triple product a·(b×c) and its geometric interpretation in 3D space
Input Vectors
Vector a
Vector b
Vector c
Results
Step-by-Step Solution:
Visualization
Scalar Triple Product Formula
Definition
The scalar triple product a·(b×c) is the dot product of vector a with the cross product of vectors b and c. It represents the volume of the parallelepiped formed by the three vectors.
Formula
Method 1: Using dot and cross products
a·(b×c) = a₁(b₂c₃ - b₃c₂) + a₂(b₃c₁ - b₁c₃) + a₃(b₁c₂ - b₂c₁)
Method 2: Using determinant
a·(b×c) = |a₁ a₂ a₃b₁ b₂ b₃c₁ c₂ c₃|
Volume Formula:
Volume of Parallelepiped = |a·(b×c)|
Properties
- a·(b×c) = b·(c×a) = c·(a×b)
- a·(b×c) = -a·(c×b) = -(c×b)·a
- a·(a×b) = 0 and a·(b×a) = 0
- If a, b, and c are coplanar (lie in the same plane), then a·(b×c) = 0
- The absolute value |a·(b×c)| represents the volume of the parallelepiped formed by vectors a, b, and c
- If vectors are represented as rows of a matrix, the scalar triple product is the determinant of that matrix
Geometric Interpretation
The scalar triple product a·(b×c) can be interpreted as:
- The volume of the parallelepiped formed by the three vectors (taking the absolute value)
- The signed volume, where the sign indicates the orientation of the vectors (positive if a, b, and c form a right-handed system)
- The projection of vector a onto the normal vector of the plane defined by b and c, multiplied by the area of the parallelogram formed by b and c
Applications of Scalar Triple Product
Physics
The scalar triple product is used in calculating volumes in physics problems, such as the work done by a force in a three-dimensional field. In electricity and magnetism, it appears in formulas for the electric field due to a magnetic dipole.
Engineering
In mechanical and civil engineering, the scalar triple product helps calculate moments of inertia, torque, and the volume of complex three-dimensional structures. It's also used in fluid dynamics to calculate flow rates and circulation.
Computer Graphics
In 3D graphics and computational geometry, the scalar triple product helps determine if a point is inside a tetrahedron, calculate volumes of irregular shapes, and detect coplanarity of points in space.
Mathematics
In linear algebra, the scalar triple product is related to the determinant of a 3×3 matrix. In vector calculus, it appears in formulas involving the divergence of a curl (which is always zero). In differential geometry, it's used to compute curvature and torsion of space curves.
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
a·(b×c) = signed volume.
— Scalar Triple
Zero when a, b, c are coplanar.
— Coplanarity
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