## Algebra of Vectors

2.3 Algebra of Vectors

### Learning Objectives

By the end of this section, you will be able to:

- Apply analytical methods of vector algebra to find resultant vectors and to solve vector equations for unknown vectors.
- Interpret physical situations in terms of vector expressions.

### Vector Algebra

Vectors can be added together and multiplied by scalars.

Resolving vectors into their scalar components (i.e., finding their scalar components) and expressing them analytically in vector component form allows us to use vector algebra to find sums or differences of many vectors *analytically* (i.e., without using graphical methods). The resultant of adding two vectors and is:

**Discuss!**

Work through this problem on your own. Then bring your work to class for a group discussion.

Three displacement vectors ** A**,

**, and**

*B***, are specified by their magnitudes**

*F**A*= 10.00,

*B*= 7.00, and

*F*= 20.00, respectively, and by their respective direction angles with the horizontal direction α = 35°, β = −110°, and φ = 110°. The physical units of the magnitudes are centimeters.

Use the analytical method to find vector G, which is:

Verify that *G* = 28.15 cm and that 𝜃 = −68.65°.

### Unit Vectors

A unit vector is a dimensionless entity—that is, it has no physical units associated with it. The general rule of finding the unit vector of direction for any vector is to divide it by its magnitude *V*: