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

MA 227 Standard S02


The Dot Product

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

  • S02: DotProd. Compute and apply the dot product of two vectors.

S02. The Dot Product

  • The dot product \(\vect v\cdot\vect w\) measures the work done by a force given by \(\vect v\) along a displacement given by \(\vect w\).
    • If the force is in the same direction of the displacement, then work is simply given by scalar multiplication of the magnitudes of \(\vect v,\vect w\).
    • If the force is orthogonal to displacement, then no work is done.
    • It may be computed by \(\vect v\cdot\vect w=\|\vect v\|\|\vect w\|\cos\theta\) where \(\theta\) is the angle between the vectors.
  • The following properties may be proven about the dot product.
    • \(\vect{v} \cdot \vect{w} = \vect{w}\cdot\vect{v}\)
    • \((c\vect{v})\cdot \vect{w} = \vect{v} \cdot (c\vect{w}) = c(\vect{v} \cdot \vect{w})\)
    • \(\vect{v} \cdot (\vect{w}_1 + \vect{w}_2) = \vect{v}\cdot\vect{w}_1 + \vect{v}\cdot \vect{w}_2\)
    • \(\vect{v} \cdot \vect{v} = \|\vect{v}\|^2\)
    • \(\vect{0} \cdot \vect{v} = 0\)
  • These properties may be used to obtain a simplified formula: \(\vect v\cdot\vect w=v_xw_x+v_yw_y+v_zw_z\).
    • This allows \(\theta\) be easily computed from \(\cos\theta=\frac{\vect v\cdot\vect w}{\|\vect v\|\|\vect w\|}\).
    • Thus two nonzero vectors are orthogonal if and only if \(\vect v\cdot\vect w=0\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 11.3 (exercises 1-8[parts a,b], 31-32, 41)