Multivariable Calculus (Calc III) - Complete Semester Course - Vector Cross Product, Multivariable Calculus

(Unit 1 Lecture 10) The vector cross product in R3--how to compute it, algebraic properties, relationship to orthogonality, the right-hand rule, and a formula relating ||axb|| to the sine of the angle between a and b. In detail: in this lecture, we learn the cross product, a vector operation in three-dimensional space, ℝ3. Unlike the dot product, which results in a scalar, the cross product of two vectors in ℝ3 produces a third vector also in ℝ3. This operation is vital in various fields, particularly in physics for understanding concepts like torque. We learn methods of computing the cross product, its algebraic properties, and its geometric interpretation, emphasizing the right-hand rule and the significance of the cross product's magnitude. 1. Cross Product Definition: The cross product is applicable only in ℝ3, producing a vector orthogonal to the two vectors being crossed. 2. Computational Method: The cross product of vectors 𝐮 and 𝐯 can be computed using a matrix-like structure with the standard basis vectors 𝐢,𝐣,𝐤 and the components of 𝐮 and 𝐯. 3. Algebraic Properties: - Non-Commutative: 𝐮×𝐯≠𝐯×𝐮. In fact, 𝐮×𝐯=−𝐯×𝐮. - Distributive over vector addition. - Scalar Multiplication: Compatible with scalar multiplication before or after the cross product. 4. Geometric Interpretation: The resultant vector from a cross product is perpendicular to both original vectors, determined using the right-hand rule. 5. Magnitude of Cross Product: The length of 𝐚×𝐛 is ||𝐚||⋅||𝐛||⋅sin(𝜃), where 𝜃 is the angle between 𝐚 and 𝐛. 6. Parallel Vectors: If 𝐚×𝐛=0, then 𝐚 and 𝐛 are parallel (or at least one is the zero vector). 7. Applications: We will see that the cross product is crucial in physics for calculating torque and in vector calculus for understanding spatial relationships in ℝ3. #mathematics #math #multivariablecalculus #calculus #vectorcalculus #iitjammathematics #calculus3 #crossproduct