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:
The answer would be "j"
Step-by-step explanation:
First, you would want to find the value of the original equation:
5(y + 2) + 4
Use order of operations and distribute the five.
5y + 10 + 4
5y +14
This is the value of the original equation
Now we can work through the other options, but because we already know the answer, lets see about that one.
5 x y + 5 x 2 + 4
Again, use order of operations.
Multiply first
5y + 10 + 4
and complete the equation
5y + 14, which equals our original equation.
Answer: y=-8
Step-by-step explanation:
If the slope is zero that means that the line does not go up (or have any rise) and therefore must be a horizontal line. Since it passes through the point (3,-8), the equation has to be y=-8.
Answer:
13÷ 23
Step-by-step explanation:
The computation of the odds of player winning is shown below;
For $5 it is 40%
And, for $100 it is 25%
so for total it is
= 40% + 25%
= 65%
And, the total percentage is 100%
So, the odds in favor is
65:100
13:20
Now for winning it would be 13÷ 23
