Figure 1 Coordinate system

The laws of sine

We use Eq.2.8 to find sides: a, b or c, Figure 2.21.

We use Eq.2.9 to find angles: A, B, C, shown in Figure 2.21.
The laws of cosine



We use Eq. 2.10 to find sides.



Eq.2.11 helps to find angles.
Unit vectors Unit vectors describe directions in space. A unit vector has a magnitude of one, with no units. The unit vectors in Cartesian coordinate system are denoted by and . Unit vectors are always aligned with the x-,y-, and z-axes of a Cartesian coordinate system.

Scalar product The scalar product (or dot product) of the vectors and is given by

The scalar product obeys the commutative law of multiplication; the order of the two vectors does not matter. If two vectors are perpendicular then their scalar product is zero.
Vector product The vector product of two vectors and , also called the cross product, is

The vector product is anti–commutative:
The vector product of two vectors is also found by using the components. The components of are given


The vector product can be abbreviated by the notation of the determinant:



Questions

1. 
   How many components are required to describe vectors in two dimensions? How many components does a vector in 3D have?
   
2. 
   Can a vector have a component greater than its magnitude?
   
3. 
   How can we find the resultant vector of two or more vectors?
   
4. 
   Can a vector has a component equal to zero and still has a nonzero magnitude?
   
5. 
   Under what circumstances would a vector have components that are equal in magnitude?
   
6. 
   Two vectors have unequal magnitudes. Can their sum be zero?
   
7. 
   Is it possible to add a vector quantity to a scalar quantity?
   
8. 
   Explain unit vectors in scalar product.
   
9. 
   Multiplication of unit vectors in vector product.
   
10. 
    Is the scalar product of equal to ? What can we say about the vector product of and ?
    

Problems

1. 
   A bicyclist rides around a circle of diameter 20 m. (a) Find the magnitude of displacement vector if bicyclist traveled halfway of the circle. (b) How far does this person move? (c) What will the magnitude of the displacement be if the circle is completed? (Ans: (a) 20 m; (b) 31.4 m; (c) 0).
   
2.