Question

9 - Determine o ponto de interseção, caso exista, entre as retas de equações paramétricas: x = 7+3s, y = -4-3s, z = -7-5s e x = -5+6 t, y = 1+t, z = 5-2t​

Answer 1

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$\sf r:\begin{cases}\sf x=7+3s\\\sf y=-4-3s\\\sf z=-7-5s\end{cases}\\\sf s:\begin{cases}\sf x=-5+6t\\\sf y=1+t\\\sf z=5-2t\end{cases}\\\sf y=y\implies -4-3s=1+t\\\sf t+3s=-4-1\\\sf t+3s=-5~\boxed{\sf1} \\\sf z=z\implies -2t+5=-7-5s\\\sf -2t+5s=-7-5\\\sf -2t+5s=-12~\boxed{\sf2}\\\underline{\sf de~\boxed{\sf1}~e~\boxed{\sf2}~vem:}$

$\begin{cases}\sf t+3s=-5\cdot(2)\\\sf-2t+5s=-12\end{cases}\\+\underline{\begin{cases}\sf\diagdown\!\!\!\!\!\!2t+6s=-10\\\sf-\diagdown\!\!\!\!\!2t+5s=-12\end{cases}}\\\sf 11s=-22\\\sf s=-\dfrac{22}{11}\\\sf s=-2\\\sf t+3s=-5\implies t=-5-3s\\\sf t=-5-3\cdot(-2)\\\sf t=-5+6\\\sf t=1\\\underline{\sf vamos~substituir~t~na~reta~s:}\\\boxed{\sf x=-5+6\cdot1=1\checkmark}\\\boxed{\sf y=1+1=2\checkmark}\\\boxed{\sf z=5-2\cdot1=3\checkmark}\\\huge\boxed{\boxed{\boxed{\boxed{\sf r\cap s:=\{1,2,3\}\checkmark}}}}$