PLEASE QUICK HELP ME! WHATS THE ANSWER

Solved

Show answers

Ответ:

noah12345678

07.10.2022

Mathematics

Extract the normal vectors from the given planes:

$-2x+y+z+2=0\implies\vec n_1=(-2,1,1)$

$x+y-3z+1=0\implies\vec n_2=(1,1,-3)$

(which are unique up to their signs, meaning either $\vec n_1$ or $-\vec n_1$ are valid choices for the normal vector)

The third plane must be perpendicular to both these given planes, which means it would be parallel to both $\vec n_1$ and $\vec n_2$ , which in turn means its own normal vector $\vec n_3$ should be perpendicular to both $\vec n_1$ and $\vec n_2$ .

Enter the cross product:

$\vec n_3=\vec n_1\times\vec n_2=(-4,-5,-3)$

or (4, 5, 3), which also works.

The given plane passes through (-1, 1, 4), so its equation is

$(x+1,y-1,z-4)\cdot\vec n_3=0$

Simplify:

$(x+1,y-1,z-4)\cdot(4,5,3)=0$

4(x+1)+5(y-1)+3(z-4)=0

$\boxed{4x+5y+3z=13}$

0,0(0 оценок)

Ask your question to the AI advisor

Ai-bot is an expert in any field and is the perfect companion for reliable and useful answers and advice on a variety of topics, including science, history, technology, art, sports, health, culture and more.

Ask your question

More tips

Answers on questions: Mathematics

Ask an AI advisor a question

I want advice