# 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 .

Arun Kumar IIT Delhi
8 years ago
Equation of plane with three points
 x-x1 y-y1 z-z1 = 0 x2-x1 y2-y1 z2-z1 x3-x1 y3-y1 z3-z1

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

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

 7 x + 10 y - 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
=>$\begin{pmatrix} x &y &z \\ -1& 2 &0 \\ 1&1 &4 \end{pmatrix}$
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 =
$5\sqrt{14}/7$

divide by velocity to get the answer

Arun Kumar
IIT Delhi
Arun Kumar IIT Delhi
8 years ago
Equation of plane with three points
$\begin{pmatrix} x-x_{1} &y-y_{1} &z-z_{1} \\ x_{2}-x_{1}&y_{2} -y_{1} & z_{2}- z_{1} \\ x_{3}-x_{1}&y_{3} -y_{1} & z_{3}- z_{1} \end{pmatrix} \\ x_{1}=4,-1,5\\ x_{2}=-1,2,0\\ x_{3}=1,1,4\\ solving \\7x-10y-z=13$

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
=>$\begin{pmatrix} x &y &z \\ -1& 2 &0 \\ 1&1 &4 \end{pmatrix}$
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 =
$5\sqrt{14}/7$

divide by velocity to get the answer

Arun Kumar
IIT Delhi