From: Andrew Lorimer Date: Tue, 6 Nov 2018 10:08:00 +0000 (+1100) Subject: spec - finalise vectors stuff X-Git-Tag: yr11~6 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/f0298826faf08bacfdc5af1e9fdd7d4ba5e8eaa0 spec - finalise vectors stuff --- diff --git a/spec/graphics/parallelogram-vectors.jpg b/spec/graphics/parallelogram-vectors.jpg new file mode 100755 index 0000000..8bafae1 Binary files /dev/null and b/spec/graphics/parallelogram-vectors.jpg differ diff --git a/spec/graphics/vector-subtraction.jpg b/spec/graphics/vector-subtraction.jpg new file mode 100755 index 0000000..f94832b Binary files /dev/null and b/spec/graphics/vector-subtraction.jpg differ diff --git a/spec/vectors.md b/spec/vectors.md index dcdf2c5..2374b2a 100644 --- a/spec/vectors.md +++ b/spec/vectors.md @@ -5,6 +5,10 @@ graphics: yes tables: yes author: Andrew Lorimer classoption: twocolumn +header-includes: +- \usepackage{harpoon} +- \usepackage{amsmath} +- \pagenumbering{gobble} --- @@ -13,7 +17,7 @@ classoption: twocolumn - **vector:** a directed line segment - arrow indicates direction - length indicates magnitude -- notated as $\vec{a}, \widetilde{A}$ +- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$ - column notation: $\begin{bmatrix} x \\ y \end{bmatrix}$ @@ -47,13 +51,18 @@ Parallel vectors have same direction or opposite direction. Vectors may describe a position relative to $O$. -For a point $A$, the position vector is $\boldsymbol{OA}$ +For a point $A$, the position vector is $\overrightharp{OA}$ + +\vfill\eject ## Linear combinations of non-parallel vectors If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then: -$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad\text{implies}\quad m = p, \> n = q$$ +$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q$$ + +![](graphics/parallelogram-vectors.jpg){#id .class width=20%} +![](graphics/vector-subtraction.jpg){#id .class width=10%} ## Column vector notation @@ -95,11 +104,10 @@ $$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$ 3. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$ For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$: -$\boldsymbol{a \cdot b}=\{ - \begin{array}{ll} - |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction} \\ - -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions} \\ - \end{array}$ +$$\boldsymbol{a \cdot b}=\begin{cases} +|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\ +-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions} +\end{cases}$$ ## Geometric scalar products diff --git a/spec/vectors.pdf b/spec/vectors.pdf index b60fffc..c468f7f 100644 Binary files a/spec/vectors.pdf and b/spec/vectors.pdf differ