1. We live in a Three Dimensional World
Vectors in Space
We live in a Three Dimensional World

2. Rectangular Coordinates in Space
Right handed coordinate system
We now have an ordered triple (x,y,z) associated with each point.
Graphing examples

3. Representing Vectors in Space
Since we have a new axes, we will now need a third unit vector to represent the z axis.
i = (1, 0, 0) j = (0, 1, 0) and
k = (0, 0, 1)

4. Position Vector To find the position vector, we will now have
v = (a2 – a1)i + (b2 – b1)j + (c2 – c1)k

5. Addition, Subtraction and Scalar Multiplication
All rules that applied in two dimensions, now apply in three dimensions

6. Unit Vector in Direction of v
For any non zero vector v, the vector
is a unit vector that has the same direction as v.

7. Dot Product
We find the dot product the same way we found it in two dimensions, we just add the third dimension

8. Angle Between Two Vectors
We use the same formula we used in two dimensions including the third dimension

9. Direction Angles of Vectors in Space
This is the only truly new operation.
There are three direction angles
a = angle between v and the positive x-axis, 0 ≤ a ≤ p
b = angle between v and the positive y-axis, 0 ≤ b ≤ p
g = angle between v and the positive z-axis, 0 ≤ g ≤ p

10. Direction Angles

11. Direction Cosines
The direction cosines play the same role in space as slope does in the plane.

12. Property of Direction Cosines
If a, b, and g are the direction angles of a nonzero vector v in space, then
cos2 a + cos2b + cos2 g = 1