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.
Answer:
Dream Destination Hawaii Number of Nights 4 Expense Total budgeted for each expense for the entire trip Round-Trip Airfare Airline 5,000 Cost per ticket dont know (you can use orbitz.com, kayak.com, or expedia.com) Lodging (Circle one) Hotel or Airbnb Cost per night dont know (you can check hotel prices on hotels.com and short-term vacation rentals on airbnb.com) Transportation (circle one) Car rental is $45 per day total Uber/taxis are $30 per day total Subway/buses are $10 per day total Food he (write your best estimate) Miscellaneous (museums, concerts, tips for servers, souvenirs, etc.) we (write your best estimate) Total $ ,780 Are you within your budget? yes If not, reduce one expense or the number of nights you’re staying above.
Step-by-step explanation:
Hi,
850 $ -- 6%*850 $ = 850 $ -- 6*850/100 $ = 850 $ -- 5100/100 $ = 850 $ -- 51 $ = 799 $.
Easy, huh ?
Green eyes.
Answer:
(B) 1.2 > -6.9
Explanation:
The negative numbers/decimals are always to the left of the 0 on a number line. They are not positive.
The positive numbers are always to the right of the 0 on a number line. They are always greater than negative numbers.
6.9 might be greater than 1.2 if it is a positive number, but if it is a negative number, 1.2 is greater.
