Multivariable Calculus (Calc III) - Complete Semester Course - Cross Product Areas and Volumes, Multivariable Calculus

(Unit 1 Lecture 11) We learn how to compute the area of a triangle or parallelogram, and how to compute the volume of a parallelepiped. Key Points 1. Cross Product in ℝ3: The cross product of vectors 𝑎⃗ and 𝑏⃗ yields a vector orthogonal to both, with a magnitude of ‖𝑎⃗ ‖‖𝑏⃗ ‖sin(𝜃). 2. Area of Parallelogram: The area is given by the magnitude of the cross product of the vectors forming its sides. 3. Triangle Area in ℝ2: For a triangle with vertices 𝑃, 𝑄, and 𝑅, the area is half the magnitude of the cross product of vectors PQ and PR. 4. Volume of Parallelepiped: The volume is the absolute value of the scalar triple product 𝑐⃗ ⋅(𝑎⃗ ×𝑏⃗ ). #mathematics #math #multivariablecalculus #calculus #vectorcalculus #iitjammathematics #calculus3 #crossproduct