Every covariance matrix entry measures how much two assets move together; the cross product answers a complementary question — what lies perpendicular to two vectors? That normal direction is the foundation for surface integrals, orientation of option payoff surfaces, and the signed volume formula for the triple scalar product, which reappears later as the Jacobian determinant in multivariate density transformations. Here you build the cross product from its determinant formula, establish its core geometric properties (direction via the right-hand rule, magnitude as the area of a parallelogram), and extend to the triple scalar product as a signed volume.
Learning objectives. By the end of this note you should be able to:
- compute from the determinant expansion;
- verify that is perpendicular to both and ;
- state the magnitude formula and use it to compute areas of triangles and parallelograms;
- apply the right-hand rule to determine the direction of ;
- compute the triple scalar product as a determinant and interpret it as a signed volume;
- identify when three vectors are coplanar using the zero-determinant criterion.
In this lesson
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- Definition and the Determinant Formula
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- Perpendicularity — the Core Property
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- The Right-Hand Rule and Geometric Magnitude
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- Area of Triangles and Parallelograms
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- The Triple Scalar Product and Volume
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- Summary Table
Before you start
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