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 .

Question

Arun Kumar · Answer

Equation of plane with three points x - x 1 y - y 1 z - z 1 = 0 x 2 - x 1 y 2 - y 1 z 2 - z 1 x 3 - x 1 y 3 - y 1 z 3 - z 1 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 ={ a x - b x ; a y - b y ; a z - b z }={(-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

Arun Kumar · Answer

Equation of plane with three points a - b ={ a x - b x ; a y - b y ; a z - b z }={(-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

x-x₁	y-y₁	z-z₁	= 0
x₂-x₁	y₂-y₁	z₂-z₁
x₃-x₁	y₃-y₁	z₃-z₁

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

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