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)`

P (vecp) and Q (vecq) are the position vectors of two fixed points and R(vecr) is the postion vector of a variable point. If R moves such that (vecr-vecp)xx (vecr-vecq)=vec0 then the locus of R is

P(vec p) and Q(vec q) are the position vectors of two fixed points and R(vec r) is the position vectorvariable point. If R moves such that (vec r-vec p)xx(vec r -vec q)=0 then the locus of R is

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

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. Radius of the sphere, with (2, -3, 4) and (-5, 6, -7) as xtremities of a diameter, is

If vec aa n d vec b are non-collinear vectors, find the value of x for which the vectors vecalpha=(2x+1) vec a- vec b""a n d"" vecbeta=(x-2)"" vec a+ vec b are collinear.

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 x vec a+y vec b+z vec c=0a n dx+y+z=0, then show that A ,Ba n dC lie on a line.

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