Find the equation of the plane through the points `A(2,1,-1)` and `B(-1,3,4)` and perpendicular to the plane `x-2y+4z=10`. Also, show that the plane thus obtained contains the line `vecr=(-hati+3hatj+4hatk)+lambda(3hati-2hatj-5hatk)`.

Find the equation of the plane through the points `A(2,1,-1)` and `B(-

1.	Find the equation of the plane through the points `A(2,1,-1)` and `B(-1,3,4)` and perpendicular to the plane `x-2y+4z=10`. Also, show that the plane thus obtained contains the line `vecr=(-hati+3hatj+4hatk)+lambda(3hati-2hatj-5hatk)`.
Answer» Correct Answer - `18x+17y+4z=49` Obtain the required equation of the plane as `18x+17y+4z=49`………(A) The given line is `vecr=(3lambda-1)hati+(3-2lambda)hatj+(4-5lambda)hatk`. Coordinates of any points on this line are `(3lambda-1,3-2lambda,4-5lambda)`. These coordinates satisfy (A), as shown below: LHS =`18(3lambda-1)+17(3-2lambda)+4(4-5lambda)=49` =RHS. So, `(3lambda-1,3-2lambda,4-5lambda)` lies on plane (A). Hence, the plane (A) obtained above equation the given line.

Find the equation of a line passing through the point `(2hati-3hatj-5hatk)` and perpendicular to the plane`vecr.(6hati-3hatj+5hatk) +2=0`. Also find the point of intersection of this line and the plane.
If the points `(1,1,p)` and `(-3,0,1)` be equidistant from the plane `vecr.(3hati+4hatj-12hatk)+13`, find the values of p.
Find the distance of the point (1,2,5) from the plane `vecr.(hati+hatj+hatk)+17=0`
Find the co-ordinates of the foot of perpendicular and the length of perpendicular drawn from the point `(2,3,7)` to the plane `3x-y-z=7`.
Find the length of the foot of the perpendicular from the point (1,1,2) to the plane `vecr.(2hati-2hatj+4hatk)+5=0`
Find the coordinates of the image of the point P`(1, 3, 4)` in the plane `2x-y+z+3=0`.
Find the equationof the plane passing through each group of points. i) A(2,2,-1), B(3,4,2) and C(7,0,6) ii) A(0,-1,-1), B(4,5,1) and C(3,9,4) iii) A(-2,6,-6), B(-3,10,-9) and C(-5,0,-6).
Find the coordinates of the foot of the perpendicular and the perpendicular distance from the point P(3,2,1) to the plane `2x-y+z+1=0`.
Find the distance of the point `(-1,-5,-10)` from the point of the intersection of the line `vecr = 2hati-2hatk+lambda(3hati+4hatj+2hatk)` and the plane `vecr.(hati-hatj+hatk) = 5`.
Find the distance of the point `(2hati-hatj-4hatk)` from the plane `vecr.(3hati-4hatj+12hatk)=9`.