From 78674466008832e9a97c8adc19c0e7e868d86ec7 Mon Sep 17 00:00:00 2001 From: Mars Ultor Date: Thu, 18 Sep 2025 01:13:31 -0500 Subject: [PATCH] i'm slowing down. problem 10 of the fourth lab done --- labs-set-1/template.pdf | Bin 161709 -> 168359 bytes labs-set-1/template.tex | 38 ++++++++++++++++++++++++++++++++++++++ 2 files changed, 38 insertions(+) diff --git a/labs-set-1/template.pdf b/labs-set-1/template.pdf index fc4ca0deb28f2ce8c87e334e69fd7af5855ea891..5b119066efcc03428c17e8532533a75b363a6814 100644 GIT binary patch delta 51369 zcmZ4co^$zPt_=q>S&Yq0%qAzYXw=`0@y?eq7uxqa{7Yd;#;lZ&k_nA>a~?5R9+hUS z-cT%%5-Aqe@8b1f|NeN>**V`U;Qi-$!P0KYY(FTl-_r$;AcF{lD%uUwm+@p^`(&pUo*>Y^G=3G2577+;3jl zp>Owzn^Enlg7%tCy+JB-`l?S~+MKCv=Q&Sjl95;C2Swp62Ul;MHcP#r^6T%j?e%f7 zpFGUZdwM8u2{knN{!-a$Q8dr0ou^rJOzfO<1XV5TFKeuAnXENsU2DPBIWrSjBb!o1 zUdaaU{qU#mh4g{2sN_|QSy9s#CC@xrd0_Xf8Ar8}OEzmImE3aqy-&N|<@=q?3x7XY zn!B#=KYO!1Q7~RBfOE0U%BaqnCP7aec$Bof_gy^1n$E#>u!uD|D>v=bEGy%kv-dL3 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zUC_9jwElkmeTf@}Y{ADq>@oawMB;yj&p*RuvwLT*KXynz`RBxqn{wCKeO@cE`9bor z3oLDF1||pFk3DZS5uGGAg;*z52 UtrMBVxeY9gjJZ@*UH#p-0Pq8Eb^rhX diff --git a/labs-set-1/template.tex b/labs-set-1/template.tex index 5ab4680..4052c60 100644 --- a/labs-set-1/template.tex +++ b/labs-set-1/template.tex @@ -293,4 +293,42 @@ \end{align*} } +\qs{}{ + \begin{align*} + \vec{u} &= \langle -5, -3, 4 \rangle, \quad + \vec{v} = \langle 5, -4, 2 \rangle, \quad + \vec{w} = \langle 10, -5, -8 \rangle \\[2mm] + \vec{u} \times \vec{v} &= + \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + -5 & -3 & 4 \\ + 5 & -4 & 2 + \end{vmatrix} \\[1mm] + &= \hat{i}((-3)(2) - (4)(-4)) + - \hat{j}((-5)(2) - (4)(5)) + + \hat{k}((-5)(-4) - (-3)(5)) \\[1mm] + &= \hat{i}( -6 + 16 ) - \hat{j}( -10 - 20 ) + \hat{k}( 20 + 15 ) \\[1mm] + &= \langle 10, 30, 35 \rangle \\[1mm] + \vec{w} \cdot (\vec{u} \times \vec{v}) &= + \langle 10, -5, -8 \rangle \cdot \langle 10, 30, 35 \rangle \\[1mm] + &= 10\cdot10 + (-5)\cdot30 + (-8)\cdot35 \\[1mm] + &= 100 - 150 - 280 \\[1mm] + &= -330 \\ + \text{This corresponds with answer choice B} + \end{align*} +} + + +\dfn{Area of a Triangle using a Cross Product}{ + \begin{align*} + \text{Area}_{\triangle ABC} = \frac{1}{2} \|\vec{B} - \vec{A} \times \vec{C} - \vec{A}\| + \end{align*} +} + +\dfn{Area of a Parallelogram using a Cross Product}{ + \begin{align*} + \text{Area}_{\text{parallelogram}} = \|\vec{u} \times \vec{v}\| + \end{align*} +} + \end{document}