DotProductExample 2006-07-22 15:56:56 2006-07-22 15:56:56 Example dot product % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here {\bf Examples} (1) Calculate $ \mathbf{A} \cdot \mathbf{B} $ with $ \mathbf{A} = 5 \mathbf{\hat{i}} + 2 \mathbf{\hat{j}} - \mathbf{\hat{k}}$ \\ $ \mathbf{B} = 3 \mathbf{\hat{i}} - 3 \mathbf{\hat{j}} + 3\mathbf{\hat{k}}$ \\ answer: $ \mathbf{A} \cdot \mathbf{B} = (5)(3) + (2)(-3) + (-1)(+3) $ \\ $ \mathbf{A} \cdot \mathbf{B} = 15 - 6 - 3 = 6$ \\ (2) Find the angle between the above vectors. answer: We know their dot product, so we just need to calculate their magnitudes $ \left | \mathbf{A} \right | = \sqrt{A_x^2 + A_y^2 + A_z^2} = \sqrt{5^2 + 2^2 +(-1)^2} $ \\ $ \left | \mathbf{A} \right | = \sqrt{25 + 4 + 1} = \sqrt{30} = 5.48 $ \\ \\ $ \left | \mathbf{B} \right | = \sqrt{3^2 + (-3)^2 +(3)^2} $ \\ $ \left | \mathbf{B} \right | = \sqrt{9 + 9 + 9} = \sqrt{27} = 5.2 $ \\ \\ Finally $ \mathbf{A} \cdot \mathbf{B} = \left | \mathbf{A} \right | \left | \mathbf{B} \right | \cos \theta$ $ \theta = cos^{-1} \left ( \frac{ \mathbf{A} \cdot \mathbf{B} }{ \left | \mathbf{A} \right | \left | \mathbf{B} \right |} \right )$ \\ $ \theta = cos^{-1} \left ( \frac{ 6 }{ (5.48)(5.2)} \right) $ \\ $ \theta = cos^{-1} (0.21) = 77.9^o $