MAT 21C Lecture Notes - Lecture 15: Scalar Multiplication, Vector Projection, Product Rule

• School: UC-Davis
• Department: Mathematics
• Course: MAT 21C
• Professor: Naoki Saito

Document Summary

Let ~u = hu1, u2, u3i and v = hv1, v2, v3i be two vectors and k a scalar. Addition: ~u + ~v = hu1 + v1, u2 + v2, u3 + v3i. Scalar multiplication: k ~u = hku1, ku2, ku3i. Dot product: ~u ~v = u1v1 + u2v2 + u3v3. Cross product: ~u ~v = h i. Rule: vectors ~u and ~v are orthogonal if ~u ~v = 0, nonzero vectors ~u and ~v are parallel i ~u ~v = 0. cos = ~u ~v = (| ~u || ~v | sin ) ~n. The plane through p0(x0, y0, z0) normal to ~n = a~i +b ~j +c ~k has: vector equation: ~n . P0p = 0: component equation: a(x x0) + b(y y0) + c(z z0) = 0, component eqation simpli ed: ax + by + cz = d, where d = ax0 + by0 + cz0.