1. Unit Vectors
AP Physics

2. Unit Vector Unit Vector is just a fancy name for a vector of length 1.
We can construct a unit vector from any vector by simply dividing by the magnitude of the vector.
Unit vectors in the x, y, z directions are denoted:

3. In unit vector notation, vector addition is trivial.
Vector A has a magnitude of 8.94 units and points in a direction of degrees.

4. In unit vector notation, vector addition is trivial.

5. 3 D Space
We can expand the 2-dimensional (x-y) coordinate system into a 3-dimensional coordinate system, using x-, y-, and z-axes.
The x-y plane is horizontal in our diagram above and shaded green. It can also be described using the equation z = 0, since all points on that plane will have 0 for their z-value.
The x-z plane is vertical and shaded pink above. This plane can be described using the equation y=0.
The y-z plane is also vertical and shaded blue. The y-z plane can be described using the equation x=0.

6. Points in 3-D Space
In 3-dimensional space, the point (2,3,5) is graphed as:
To reach the point (2,3,5), we move 2 units along the x-axis, then 3 units in the y-direction, and then up 5 units in the z-direction.

7. Distance in 3-dimensional Space
To find the distance from one point to another in 3-dimensional space, extend Pythagoras' Theorem.
Distance from the Origin
The general point P (a, b, c) is shown on the 3D graph below. The point N is directly below P on the x-y plane.
The distance from (0,0,0) to the point P (a, b, c) is given by:

8. Distance in 3-dimensional Space
Why?
The point N (a,b,0) is shown on the graph. From Pythagoras' Theorem,
Distance NP is simply c (this is the distance up the z-axis for the point P).
Applying Pythagoras' Theorem for the triangle ONP, we have:

9. 3 D Vectors
Vector P has starts at the origin O (0, 0, 0) and ends at (2, 3, 5). We can draw vector P in three dimensional space as follows:
The magnitude of vector P is given by:

10. 3 D Vectors P
Vector Applet

11. 3 D Vectors If Vector P is referenced as:
And the magnitude of vector P is given by:
Then the angle between P and the x, y, and z axes are given by:

13. One For You