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:
Step-by-step explanation:
Given that a medical study investigated the effect of calcium and vitamin supplements on the risk of older Americans for broken bones. A total of 389 older Americans who lived at home and were in good health were studied over a three-year period. While all of the 389 people took in at least 700 milligrams of calcium and 200 units of vitamin D through their normal diet, 187 of them were given additional supplements containing 500 milligrams of calcium citrate and 70 units of vitamin D daily. Of the 187 who took additional supplements, 11 of them suffered broken bones over the three-year period. Of the 202 older Americans who did not take the additional supplement, 26 of them suffered broken bones over the study period.
Group I Group II Total
n 187 202 389
favour x 11 26 37
The fraction of older Americans who were included in the study suffered broken bones during the three-year period
=Total x/total n
=
Answer:
Solution of the given equations is (-5,2)
Step-by-step explanation:
We have been given the following system of equations:
y = (2/5)x + 4
y = 2x + 12
The solution of the system of equation can be found by finding the point of intersection of both lines on the graph. The graph of both equations is attached below.
As we can see that both the lines intersect each other at one point, and that point is (-5,2). So the solution of the given equations is (-5,2)
If D is the midpoint of GH, then GH equals 2(DH) = 16
16 = 4x - 1
4x = 17
x = 17/4
x = 4.25
The answer is :
<span>A. Always
Also </span>
<span>If
two equations have different slopes but equivalent y-intercepts, they
will have one solution and that will be the point where the y-intercept
is. If two equations have different slopes and different y-intercepts,
then there will be one solution where those two lines meet. If two
equations have the same slope but different y-intercepts, the lines will
be parallel, and there is no possible intersection point. And if two
equations have equal slopes and equal y-intercepts, these lines will
have an infinite amount of solutions, because if the equations are one
the same line, every single point on that line is a solution to the
system. </span>
