Multivariable Calculus (Calc III) - Complete Semester Course - Vector Equation of a Plane and General Form, Multivariable Calculus

(Unit 1 Lecture 15) This lecture focuses on understanding planes in three-dimensional space ℝ3. The key to defining a plane is either through three noncollinear points or a point and an orthogonal direction. Error: around 12:10, the y-intercept is 3/2 not 2/3. There are more examples in the next lesson. Key Points 1. Planes in ℝ3: Planes are visualized as infinitely extending flat surfaces, distinct from lines in ℝ2. 2. Defining a Plane: The most common ways are either (i) three noncollinear points or (ii) a point and an orthogonal direction. 3. Vector Equation of a Plane: The vector equation 𝑛⃗ ⋅(𝑟⃗ −𝑟⃗ 0)=0 uses a known point 𝑃 and a normal vector 𝑛⃗ . 4. Normal Vector via Cross Product: A normal vector to a plane can be found using the cross product of two vectors lying on the plane. Given three points, this is the way to go! 5. General Equation of a Plane: The general form 𝑎𝑥+𝑏𝑦+𝑐𝑧=𝑑 is derived from the vector equation, where 𝑎, 𝑏, and 𝑐 are components of the normal vector: 𝑛⃗ =⟨𝑎,𝑏,𝑐,⟩. #mathematics #math #multivariablecalculus #calculus #vectorcalculus #iitjammathematics #calculus3 #dotproduct