The number of integer values of for which the vectors and are coplanar is

The distance of the point from the line parallel to vector and passing through the point is

The position vector of the centre of the circle

Let , A vector in a plane of and whose projection on has the magnitude is

The position vector of a point is given by where and . Then the number of possible positions of such that is

If A,B,C are three points with position vectors and   respectively, then the points are collinear if

and be the three vectors vector in the plane of whose projection of is is given by

The position vector of a point is where and . If , then the number of possible positions of is

Let A vector in the plane of and whose projection on is , is

Let then holds for

The distance of the point from the line parallel to vector and passing through the point is

For such questions, we should know different formulas of distances for line or plane.

The value of 'a' so that the volume of parallel piped formed by and   becomes minimum is

For such questions, we should know the volume of parallelepiped.

There are following extraction process of silver but not

in terms  terms only  is