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:
There isn't one.
Step-by-step explanation:
The square root of a negative number is imaginary. There is no real number that is the square root of -3.24.
Answer:
100.... my kind sir
Step-by-step explanation:
Answer:
Probability that the piece will be filled with fudge is 4/15 = 0.2667
Step-by-step explanation:
In the question, a box of chocolate contains 15 identical looking pieces. These chocolate pieces are filled in the following way: There are
3 caramel
2 cherry cream
2 coconut
4 chocolate whip
4 fudge
So this makes 3+2+2+4+4= 15
As the total number of chocolate pieces are 15
Let E be the event that the chocolate piece is filled with fudge
As we know Probability of an event is
P(E) = number of favorable outcomes / number of possible outcomes
Here the favorable is the number of pieces filled with fudge and number of possible outcomes are 15 as there are 15 types of chocolate pieces. So the probability that the piece selected will be filled with fudge is:
P(E) = 4/15 = 0.2667