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Understanding the Cross Product and Scalar Triple Product: Geometric and Algebraic Properties

Explore the fundamental concepts of the cross product and scalar triple product, their geometric interpretations, determinant formulations, and properties in vector algebra. Learn how these operations relate to areas, volumes, and torques in three-dimensional space.

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Understanding the Cross Product and Scalar Triple Product: Geometric and Algebraic Properties

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  1. Section 13.4 The Cross Product

  2. THE CROSS PRODUCT If , then the cross product of a and b is the vector NOTES: 1. The cross product is also called the vector product. 2. The cross product a× b is defined only when a and b are three-dimensional vectors.

  3. DETERMINANTS A determinant of order 2 is defined by A determinant of order 3 can be defined in terms of second order determinants as follows:

  4. THE CROSS PRODUCT AS A DETERMINANT

  5. THEOREM The vector a× b is orthogonal to both a and b.

  6. THEOREM If θ is the angle between a and b (so 0 ≤ θ≤π), then |a × b| = |a| |b| sin θ Corollary: Two nonzero vectors a and b are parallel if and only if a × b = 0

  7. A GEOMETRIC INTERPRETATION OF THE CROSS PRODUCT The length of the cross product a × b is equal to the area of the parallelogram determined by a and b.

  8. PROPERTIES OF THE CROSS PRODUCT If a, b, and c are vectors and c is a scalar, then 1. a × b = −(b × a) 2. (ca) × b = c(a × b) = a × (cb) 3. a × (b + c) = a × b + a × c 4. (a + b) × c = a × c + b × c 5. a∙ (b × c) = (a × b) ∙ c 6. a × (b × c) = (a∙ c)b − (a ∙ b)c

  9. SCALAR TRIPLE PRODUCT The product a ∙ (b × c) is called the scalar triple product of vectors a, b, and c. NOTE: The scalar triple product can be computed as a determinant.

  10. GEOMETRIC INTERPRETATION OF THE SCALAR TRIPLE PRODUCT The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product: V = |a ∙ (b × c)|

  11. TORQUE Consider a force F acting on a rigid body at a point given by the position vector r. (For example, tightening a bolt with a wrench.) The torqueτ (relative to the origin) is defined to be the cross product of the position and force vectors. That is, τ = r× F. The magnitude of the torque is |τ| = |r × F| = |r| |F| sin θ

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