If `vecalpha` is a constant vectro and `vecgamma` is the position vector of a variable point (x,y,z), show that `(vecgamma-vecalpha) vecalpha`=0 is the equation of a plane through fixed point `vec(alpha)`

Write the position vector of the point where the line vec r= vec a+lambda vec b meets the plane vec r dot (vec n)=0.

A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle |z-c|=a , the equation of a sphere of radius is |r-c|=a , where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at (-g, -f, -h) is x^2+y^2+z^2+2gx+2fy+2hz+c=0 and its radius is sqrt(f^2+g^2+h^2-c) . Q. The centre of the sphere (x-4)(x+4)+(y-3)(y+3)+z^2=0 is

Let vec a,vec b,vec c be the position vectors of three distinct points A,B,C. If there exist scalars x, y,Z (not all zero) such that xvec a+yvec b+zvec c=0 and x+y+z=0, then show that A,B and C lie on a line.

A plane P passes through the line of intersection of the planes 2x-y+3z=2,x+y-z=1 and the point (1,0,1). What is the equation of the plane P?

let vecalpha=i+2j-k , vecbeta=2i-j+3k , vecgamma=2i+j+6k be three vectors . fi vecalpha and vecbeta are both perpendicular to the vecdelta and vecdelta.vecgamma=10, then what is the magnitude of vecdelta

A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the three coordinate axes is constant.Show that the plane passes through a fixed point.

The orthogonal projection A' of the point A with position vector (1, 2, 3) on the plane 3x-y+4z=0 is

The equation of the plane through the point (1,2,3) and parallel to the plane x+2y+5z=0 is