From baa61484fba8c0936cc28ba629475821dfd34936 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 29 Oct 2018 16:54:43 +1100 Subject: [PATCH] vectors ref --- spec/vectors.md | 30 ++++++++++++++++++++++-------- spec/vectors.pdf | Bin 0 -> 142338 bytes 2 files changed, 22 insertions(+), 8 deletions(-) create mode 100644 spec/vectors.pdf diff --git a/spec/vectors.md b/spec/vectors.md index 8fa7208..dcdf2c5 100644 --- a/spec/vectors.md +++ b/spec/vectors.md @@ -1,18 +1,19 @@ --- -header-includes: - - \documentclass{standalone} - - \usepackage{cleveref} - - \usepackage{harpoon} - - \usepackage{accent} - - \usepackage{amsmath} -... +geometry: margin=2cm + +graphics: yes +tables: yes +author: Andrew Lorimer +classoption: twocolumn + +--- # Vectors - **vector:** a directed line segment - arrow indicates direction - length indicates magnitude -- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$ +- notated as $\vec{a}, \widetilde{A}$ - column notation: $\begin{bmatrix} x \\ y \end{bmatrix}$ @@ -87,6 +88,19 @@ Produces a real number, not a vector. $$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$ +## Scalar product properties + +1. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$ +2. $\boldsymbol{a \cdot 0}=0$ +3. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$ + +For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$: +$\boldsymbol{a \cdot b}=\{ + \begin{array}{ll} + |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction} \\ + -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions} \\ + \end{array}$ + ## Geometric scalar products $$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$ diff --git a/spec/vectors.pdf b/spec/vectors.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b60fffcba5edf197205dac5e2923625bce1f7622 GIT binary patch literal 142338 zcmcG#W0Yn=lqH(BZQHhO+xpVBU73}vv~AnAZ9B8lHeYqW=~?qu&suN(O#HcldjltM z_CEInxssR!Ju?G44EfyB@EQy!GZ7P!gNZc^A0G^(9Khbf)sl#njg5)uKLZS-gq5u; zz?q0q!q(UoAOD4f`bD({hjD$pzF$DRo}ZjYt!62tk%?Ff zw0rR?XIN5FgwcAEQ*us@|5+q7S5UYZHtyV<_3wDY=%|YzRd9*nhci}?KM6*u(LcQn zW5v5! zcT&saNo2B0WT~Xic2;nDgnJn0PCH64>uY*so;d2Q-eGLz@*1kPKTln+gj>kN@Ti_ppE{iS)W{Rha;V`Ui(8l&^GzfxKP9*(-tWZFu$cqm_@&yNzK3xlj&7)D08w%LP;Y3sDvrRL}{UpR5V;fZn78olTF z=jOb|=E3-ybGBie8r-}qbnNbraX~qvzJObRQ z4;KC^c`g)&S-YyX=hb|gqIc8a>-ZdQAS7#FAfy{D&fH^v9oxx6l_3sKn&ahn zOQj4m0)dhYkpyrS3+a)NuRLxz)7L^^SL~N#uP50!V^$!Uh@2B%R#6dAM;dlBo>Opz z$+#C(Mc^-4uAyYmLHV@U{MW~0; zzmNMVHiv6TR!_V#H}E&IFXrYGP5@s)c=&J_ghN@u%W{oQ_4)F%y=L`3IIXs1mI4

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