Find an equation of the plane that contains the points p(5,−1,1),q(9,1,5),and r(8,−6,0)p(5,−1,1),q(9,1,5),and r(8,−6,0).
topjm [15]
Given plane passes through:
p(5,-1,1), q(9,1,5), r(8,-6,0)
We need to find a plane that is parallel to the plane through all three points, we form the vectors of any two sides of the triangle pqr:
pq=p-q=<5-9,-1-1,1-5>=<-4,-2,-4>
pr=p-r=<5-8,-1-6,1-0>=<-3,5,1>
The vector product pq x pr gives a vector perpendicular to both pq and pr. This vector is the normal vector of a plane passing through all three points
pq x pr
=
i j k
-4 -2 -4
-3 5 1
=<-2+20,12+4,-20-6>
=<18,16,-26>
Since the length of the normal vector does not change the direction, we simplify the normal vector as
N = <9,8,-13>
The required plane must pass through all three points.
We know that the normal vector is perpendicular to the plane through the three points, so we just need to make sure the plane passes through one of the three points, say q(9,1,5).
The equation of the required plane is therefore
Π : 9(x-9)+8(y-1)-13(z-5)=0
expand and simplify, we get the equation
Π : 9x+8y-13z=24
Check to see that the plane passes through all three points:
at p: 9(5)+8(-1)-13(1)=45-8-13=24
at q: 9(9)+8(1)-13(5)=81+9-65=24
at r: 9(8)+8(-6)-13(0)=72-48-0=24
So plane passes through all three points, as required.
X and y are books
x is fiction
y=nonifiction
x+y=36
x=12+y
12+y+y=26
2y+12=26
minus 12
2y=14
divide 2
y=7
sub
x=12+y
x=12+7
x=19
nonfiction=7
Answer: 271.0638298 is - 8,- 3
Step-by-step explanation:
-3y<2+ +18=
Your answer would be "D", because when finding the GCF of a term, you want to take the most out of it. Following the procedure of D, finding the greatest shared factor of 10 and 15 and the greatest shared power between exponent 2 and exponent 3, would allow you to find the GCF.
Answer:
She had 15 minutes left.
Step-by-step explanation:
30 - 11 = 19
19 - 4 = <em>15</em>
