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.
4
search up math vvay for more explantion also paste this in there
Answer:
x = 3
Hope this helps!
Step-by-step explanation:
ΔABC is a 45°, 45°, 90° triangle. The longest side ( 6
) can be 
while AB and CB are x. Because 6 is equal to , you can see that x is equal to 6. Both AB and CB are equal to 6. ΔBDC ( CDB ) is a right triangle ( 30°, 60°, 90° ) so, BD = x, CB = 2x, and CD = 
. CB is equal to 6 so 2x = 6, x = 3.
Answer:
Step-by-step explanation:
Given that a consumer magazine article titled "How Safe Is the Air in Airplanes" reports that the air quality, as quantified by the degree of staleness, was measured on 175 domestic flights.
The purpose was to study about the safety of all flights as a whole.
Since all flights cannot be measured within a reasonable time, a sample of 175 domestic flights were taken.
a) The population of interest here would be the measure of safety of all the airplanes.
b) Sample is the 175 domestic flights selected
c) Characteristics of interest is the safety in travelling by air in Airplanes due to air quality.
