From f0298826faf08bacfdc5af1e9fdd7d4ba5e8eaa0 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 6 Nov 2018 21:08:00 +1100 Subject: [PATCH] spec - finalise vectors stuff --- spec/graphics/parallelogram-vectors.jpg | Bin 0 -> 8124 bytes spec/graphics/vector-subtraction.jpg | Bin 0 -> 3086 bytes spec/vectors.md | 24 ++++++++++++++++-------- spec/vectors.pdf | Bin 142338 -> 163619 bytes 4 files changed, 16 insertions(+), 8 deletions(-) create mode 100755 spec/graphics/parallelogram-vectors.jpg create mode 100755 spec/graphics/vector-subtraction.jpg diff --git a/spec/graphics/parallelogram-vectors.jpg b/spec/graphics/parallelogram-vectors.jpg new file mode 100755 index 0000000000000000000000000000000000000000..8bafae191eb8c9bf2af00028ed59c62c60883526 GIT binary patch literal 8124 zcmbW62Q*w?-|x>D4AHx2BYGkvqDC-8w1_Ct!srB;IJZdG7PRcddKxKI^;AS!?gL*Y9)o+IxS0d!PR}UjXRut81wPAP@+! zzIcH1X+RaAASZ{AlTkn*5K2mlOVrFX)KpZ|ml>Jpn7KKiJlq^yT)YC}!n}Of`MJ16 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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 b60fffcba5edf197205dac5e2923625bce1f7622..c468f7f808e899ee56bb054861cc25b76569e63e 100644 GIT binary patch delta 72552 zcmeFYb#NTbmo3;WW+sc7nHemLnPo9EGg&RRn86k^Gc&U+i`imX%-G_mZ{{~|-p-qi z-Tn8)R>$qmsH&*06IFThoSP?umf;HC;L)j+zeqB%F>@hMP0tT4Bk=GfpkqPsu%;Pe zL(qYA#}Wxf(7SFJoS@|MzdnEd3On0OzE?Yt$d!tkPdEd~X(?*_$z(fT7`&GCRnAjG z!T2W`#VIC!%&H6M=T|S+yVJj((b1(=t;%+8cMg;q1&k!t)Y~a|`jb*U(V%Q?wsVW; zNe`aKyW=U=&{#WDY&BkSp84^aA( zE1%ZRJUEOk{4O58ZV}%MRjzvqyu&C$U2G{wO}01J7~0EPn-+EYZJM`quWb6T;~vF= 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