Vectors

• Find the Angle Between Two Planes from Three Points
  Use a cross product to find the second plane’s normal, then compare it with the normal to z = 0 to get the acute angle.
• How to Parametrize a Circle in 3D from Its Center and Two Points
  Given a center and two points, form a radius vector and an orthonormal basis for their plane, then parametrize the circle with sine and cosine.
• How to Prove the Dot Product Is Distributive
  The dot product distributes over vector addition because its coordinate formula is a sum of products; follow a componentwise proof and worked example.
• How to Find the Shortest Distance Between Two Parallel Lines in 3D
  For parallel lines in 3D, choose one point on each line and compute the cross product with their direction vector; its magnitude gives the shortest distance.
• What Is the Tensor Product of a Vector with Itself?
  The tensor product of a vector with itself is an element of V⊗V; in coordinates, it is the outer product with entries v_i v_j.
• Why Are Complex Numbers Numbers but 2D Vectors Usually Aren’t?
  Complex numbers are numbers because they have a consistent arithmetic structure. A 2D vector becomes one if you define suitable multiplication.