Vector equation of a plane passing through r=(1,1,−2), s=(3,0,1), p=(1,1,1)

Kameron Wang

Kameron Wang

Answered question

2022-11-16

Vector equation of a plane passing through r = ( 1 , 1 , 2 ), s = ( 3 , 0 , 1 ), p = ( 1 , 1 , 1 )
For this question I did the cross product so I found rs then rp and using those I did the cross product method to find (a,b,c). After getting a, b, c which was (−3,−6,0). I plugged it into a ( x x 0 ) + b ( y y 0 ) + c ( z z 0 ). This gave me a final equation of 3 x + 6 y = 9
The answer however is completely different where they said the equation is ( 1 , 1 , 2 ) + λ ( 2 , 1 , 3 ) + μ ( 0 , 0 , 3 )
How would I know if my answer was correct from this equation?

Answer & Explanation

kavdawg8w8

kavdawg8w8

Beginner2022-11-17Added 20 answers

If you set ( x , y , z ) = ( 1 , 1 , 2 ) + λ ( 2 , 1 , 3 ) + μ ( 0 , 0 , 3 ) = ( 1 + 2 λ , 1 λ , 2 + 3 λ + 3 μ ), then
3 x + 6 y = 3 ( 1 + 2 λ ) + 6 ( 1 λ ) = 9
The two forms essentially represent the same plane.
(Since only z depends on μ, z can be any real number.)

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