At the end of the course, each student should be able to…

**S01: 3DSpace.**Plot and analyze points and vectors in three-dimensional Euclidean space.

## S01. Points and Vectors in 3D Space

- \(xyz\) space is modeled by extending a \(z\) axis perpendicular
to the \(xy\) plane through the origin.
- Points are given as ordered triples \(P=\<x,y,z\>\).
- Distance/length in \(xyz\) space may be computed using the generalized Pythagorean Theorem: \(x^2+y^2+z^2=d^2\).

- A vector \(\vect v\) measures a direction and magnitude
in 2D or 3D space.
- Vectors are often represented by the point they point when starting from the origin: \(\vect v=\<v_x,v_y,v_z\>\).
- Since the \(xy\) plane is contained in \(xyz\) space, we assume \(\<v_x,v_y\>=\<v_x,v_y,0\>\).
- Vector addition describes the total direction and magnitude obtained by moving along the second vector after the first: \(\vect v+\vect w=\<v_x+w_x,v_y+w_y,v_z+w_z\>\).
- Multiplication of a vector by a scalar (real number) gives the vector scaled by that real number: \(c\vect v=\<cv_x,cv_y,cv_z\>\).
- The vector from point \(P\) to point \(Q\) is given by their subtraction: \(Q-P=\<q_x-p_x,q_y-p_y,q_z-p_z\>\).
- The magnitude of a vector is given by its length: \(\|\vect v\|=\sqrt{v_x^2+v_y^2+v_z^2}\).
- The direction of a vector is given by dividing out its length: \(\frac{1}{\|\vect v\|}\vect v\).
- A unit vector has magnitude \(1\).

- The standard unit vectors point in the direction of the positive coordinate
axes.
- These are \(\veci=\<1,0,0\>\), \(\vecj=\<0,1,0\>\), and \(\veck=\<0,0,1\>\).
- Vectors may be expressed as a linear combination of these: \(\vect v=v_x\veci+v_y\vecj+v_z\veck\).

- The zero vector is \(\vect 0 =\<0,0,0\>\).

### Textbook References

- University Calculus: Early Transcendentals (3rd Ed)
- 11.1 (exercises 41-46, 61-62)
- 11.2 (exercises 1-30, 41)