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Grade: 11
        a bird is at a point P coordinated (4m, -1m, 5m). the bird observes two points A and B having coordinates (-1m, 2m, 0m) and (1m, 1m, 4m) respectively. at time t=0, it starts flying in a plane of three positions with a constant speed of 5 m/s in a direction perpendicular to the straight line AB till it sees A and B collinear at time t. calculate t .
4 years ago

Answers : (2)

Arun Kumar
IIT Delhi
askIITians Faculty
256 Points
							Equation of plane with three points
x-x1y-y1z-z1= 0
x2-x1y2-y1z2-z1
x3-x1y3-y1z3-z1

x- 4y- (-1)z -5= 0
(-1) - 42 - (-1)0 - 5
1 - 41 - (-1)4 - 5

x -4y -(-1)z -5= 0
-53-5
-32-1

(x -4)(3·(-1)-(-5)·2) - (y -(-1))((-5)·(-1)-(-5)·(-3)) + (z -5)((-5)·2-3·(-3)) = 0

7(x-4)+10(y-(-1))+(-1)(z-5) = 0

7x+10y-z-13=0

a
-
b
={
ax
-
bx
;
ay
-
by
;
az
-
bz
}={(-1)-1;2-1;0-4}={-2;1;-4}
Lets a x,y,z lies on the line perpendicular to AB and in the plane

then
x-4,y+1,z-5 will also lie in the plane and perpendicular to AB where x,y,z is location of bird at any time.

dot product of these two point should be zero
=>8x+4y-3z=0

since at some time it becomes collinear with AB
=> triangle formed be A,B and that point will be zero
=>
2x-y+4z=29

also it satisfies the plane so
7x+10y-z=13

solvin all three eq.
x=15/7,y=3/7,z=44/7

distance =

divide by velocity to get the answer



Arun Kumar
IIT Delhi
Askiitians Faculty
4 years ago
Arun Kumar
IIT Delhi
askIITians Faculty
256 Points
							Equation of plane with three points

a
-
b
={
ax
-
bx
;
ay
-
by
;
az
-
bz
}={(-1)-1;2-1;0-4}={-2;1;-4}
Lets a x,y,z lies on the line perpendicular to AB and in the plane

then
x-4,y+1,z-5 will also lie in the plane and perpendicular to AB where x,y,z is location of bird at any time.

dot product of these two point should be zero
=>8x+4y-3z=0

since at some time it becomes collinear with AB
=> triangle formed be A,B and that point will be zero
=>
2x-y+4z=29

also it satisfies the plane so
7x+10y-z=13

solvin all three eq.
x=15/7,y=3/7,z=44/7

distance =

divide by velocity to get the answer



Arun Kumar
IIT Delhi
Askiitians Faculty
4 years ago
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