Grade 12 Calculus and Vectors
Chapter 8: Vector Operations and Applications

Last chapter we looked at vector addition, subtraction and multiplying by a scalar, as well as the meaning and illustration of spanning of vectors. In this chapter we will further our understanding of vectors by examining how vector multiplications (cross product) and scalar multiplications (dot products) are carried out. In addition, we will study various applications of vectors and how they may be used to simplify real life complicated problems.

Chapter Overview
- application of vectors to model and slove problems involving velocity and force
- properties, computation and application of dot products of two vectors
- properties, computation and application of cross products of two vectors
- vector and scalar projections of vectors
- cross product of two algebraic vectors in three-dimensional space

Definitions worth remembering
-   Resultant (Total) Vectors:
Resultant vector (or resultant of multiple vectors), is a single vector that can be used to replace the total (overall) effect of multiple vectors. Many instructors may also refer to it as the compnents sum of all vectors acting on an object.

-   Equilibrant Vectors:
Equilibrant vector is a single vector that opposes the resultant or total vector. As such, when a resultant vector and an equilibrant vectors are applied to an object, the system of vectors obtain equilibrium.

Let R = Resultant vector
Let E = Equilibrant vector
Then,
E = - R
R + E = R + ( - R ) = R - R = 0    ← so the result is a zero vector.

8.1 Resolving Vectors



8.2 Resultant (Total) Vectors - Problems involving Velocity and Forces



8.3 Dot (Scalar) Product of Vectors



8.4 Distributive Property of Dot Products



8.5 Angle between Two Vectors



8.6 Scalar and Vector Projections



8.7 Cross (Vector) Product of Vectors



8.8 Applications of Dot Products



8.9 Applications of Cross Product