Question

find the point of intersection

Answer 1

\[x-2y+z=5 , 2x-y+z=1, -2x+y+z=3\]

Answer 2

of the three planes

Answer 3

@LastDayWork

Answer 4

We are given three equations and three unknowns ; simply solve them to get (x,y,z)

Answer 5

how to solve them I added them subtracted them I didn't get the ans.

Answer 6

Have you studied Determinants and Matrices ?

Answer 7

Determinants yes

Answer 8

but only the cross product determinanats

Answer 9

Then you must have studied Cramer's Rule, right ?

Answer 10

nope

Answer 11

Never mind ;
The equations are
x - 2y + z = 5 (1)
2x - y + z = 1 (2)
-2x + y + z = 3 (3)

Answer 12

Can you find the value of z using (2) and (3) ??

Answer 13

z=2

Answer 14

correct

Answer 15

Now, use any two equations to find the value of x and y.

Answer 16

but the ans is (-8/3,-10/3,1)

Answer 17

z=1 but how?

Answer 18

z = 1 will give 0 = -3 from (2) and (3)
Hence, book's answer is wrong.

Or you made a mistake while copying the equations.

Answer 19

I copied the equation correctly I used the Cramer's rule i got \[(\frac{ -5 }{ 3},\frac{ -7 }{ 3 },2)\]

Answer 20

But, is that correct way to do this prob since they r asking intersection point for the planes. So, shouldn't we use the normal vector and cross product stufff to find the point?

Answer 21

Any point lying on a plane satisfies the equation of the plane ; and vice-versa

Do you agree with this ^^ statement ??

Answer 22

http://www.wolframalpha.com/input/?i=x%E2%88%922y%2Bz%3D5%2C2x%E2%88%92y%2Bz%3D1%2C%E2%88%922x%2By%2Bz%3D3

Answer 23

so my answer is correct ?

Answer 24

Yes.

Answer 25

my professor is wrong then :-o

Answer 26

What did they say?

Answer 27

(-8/3,-10/3,1) professor directly gave ans

Answer 28

2nd) two lines intersect at what point x1(t)=<1,1,3>+t<-1,0,2> and x2(t)=<-1,1,4>+s<2,0,1> (for this one also I got against prof. ans.) I don't know if this is my mistake or...

Answer 29

Why is there an \(s\) in \(x_2\)?

Answer 30

Set them equal, solve for \(t\), then plug \(t\) back into either one to get the exact point.

Answer 31

so, and prof. said t and s should be equal and should satisfy both equations