410

a)
$A=\left(-53{,}\ 18{,}\ 71\right)$
$OA=OP+t\overline{v}$
$P=\left(10{,}-27{,}7\right)$
$\overline{v}=7\overline{\text{i}}-5\overline{\text{j}}-8\overline{\text{k}}$
$-53\overline{\text{i}}+18\overline{\text{j}}+71\overline{\text{k}}=10\overline{\text{i}}-27\overline{\text{j}}+7\overline{\text{k}}+7t\overline{\text{i}}-5t\overline{\text{j}}-8t\overline{\text{k}}$

$\begin{cases} -53=7t+10&t=-9\\ 18=-5t-27&t=-9\\ 71=-8t+7&t=-8 \end{cases}$

piste A ei ole suoralla

b)

$B=\left(2{,}\ -4{,}\ 5\right)$
$C=\left(-8{,}\ 0{,}\ 17\right)$
$suoran\ suuntavektori=\overline{BC}$
$\overline{BC}=-10\overline{\text{i}}-4\overline{\text{j}}+12\overline{\text{k}}$
$\overline{OA}=\overline{OB}+t\overline{v}$
$-53\overline{\text{i}}+18\overline{\text{j}}+71\overline{\text{k}}=2\overline{\text{i}}-4\overline{\text{j}}+5\overline{\text{k}}-10t\overline{\text{i}}-4t\overline{\text{j}}+12t\overline{\text{k}}$

$\begin{cases} -53=2-10t&t=5{,}5\\ 18=-4-4t&t=5{,}5\\ 71=5+12t&t=5{,}5 \end{cases}$

Piste A on suoralla