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```
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 .

```
6 years ago Arun Kumar
IIT Delhi
256 Points
```							Equation of plane with three pointsx-x1y-y1z-z1= 0x2-x1y2-y1z2-z1x3-x1y3-y1z3-z1x- 4y- (-1)z -5= 0(-1) - 42 - (-1)0 - 51 - 41 - (-1)4 - 5x -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)) = 07(x-4)+10(y-(-1))+(-1)(z-5) = 07x+10y-z-13=0a-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 planethenx-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=0since at some time it becomes collinear with AB=> triangle formed be A,B and that point will be zero=>2x-y+4z=29also it satisfies the plane so7x+10y-z=13solvin all three eq.x=15/7,y=3/7,z=44/7distance =divide by velocity to get the answerArun KumarIIT DelhiAskiitians Faculty
```
6 years ago Arun Kumar
IIT Delhi
256 Points
```							Equation of plane with three pointsa-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 planethenx-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=0since at some time it becomes collinear with AB=> triangle formed be A,B and that point will be zero=>2x-y+4z=29also it satisfies the plane so7x+10y-z=13solvin all three eq.x=15/7,y=3/7,z=44/7distance =divide by velocity to get the answerArun KumarIIT DelhiAskiitians Faculty
```
6 years ago
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