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.
Given data
<span>sin (x+pi/2)=cos x
</span>now using sin law
sin(a+b)=sin(a)cos(b)+cos(a)sin(b)
now using above values
sin(pi/2+x)=sin(pi/2)cos(x)+cos(pi/2)sin(x)
as we know that
sin(pi/2)=1
cos(pi/2)=0
now putting these values
sin(pi/2+x)=1*cosx+0*1
sin(pi/2+x)=cosx
hence proved that
<span>sin (x+pi/2)=cos x</span>
Answer:
x-intercept (-3,0); y-intercept (0,12)
Step-by-step explanation:
To find the x-intercept --> y should equal 0
8x-2(0)=-24 substitute
8x=-24 divide
x=-3 (-3,0)
To find they-intercept --> x should equal 0
8(0)-2y=-24 substitute
-2y=-24 divide
y=12 (0,12)
<span>rational number between 5.2 and 5.5 is 53/10 (cause 53/10 = 5.3)
</span>irrational number between 5.2 and 5.5 is √29 (cause <span>√29 = 5.385164....)</span>
Answer:
750 mm, 7.5 cm
Step-by-step explanation:
Verify each case
case a) 2.5 km, 2,500 m
we know that
1 km ------> 1.000 m
so
2.5 km=2,500 m
therefore
The pair of measurements is equivalent
case b) 750 mm, 7.5 cm
we know that
1 cm ------> 10 mm m
so
7.5 cm=75 mm
therefore
<u>The pair of measurements is not equivalent</u>
case c) 3 m, 3,000 mm
we know that
1 m ------> 1.000 mm
so
3 m=3,000 mm
therefore
The pair of measurements is equivalent
case d) 120 cm, 1.2 m
we know that
1 m ------> 100 cm
so
1.2 m=120 cm
therefore
The pair of measurements is equivalent
