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.
The eight should be in parentheses but the eight turns into a positive making the answer 4
hope this helps!
Speed (v) = distance (d) / time (t)
so either the second or third. if this is realistic deer can't run 100 miles per hour so is say the second initially but to check what is .25miles/25s convert to hours. (3600 seconds in an hour) so .25*3600=900 then 900miles/25hour= 36miles/ hour.
Answer:
Step-by-step explanation:
In the case of each of the answers listed above, the second derivative is equal to -4 times the function, as required by the differential equation.
For y = 2/3t^3, the second derivative is 4t, not -4y.
For y = e^(2t), the second derivative is 4y, not -4y.
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The graph shows the sum of the second derivative and 4y is zero for the answers indicated above, and not zero for the other two proposed answers.
