\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard S01

Points and Vectors in 3D Space
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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)