5. Find the vector equation of the following planes :

Solution.

Hence, the non-parametric form of the plane is

Solution.

So, the non-parametric form of the plane is

#### S N Dey mathematics class 12 solutions -Plane-Ex-5A

6. Find the vector equations of the plane passing through the points and

Solution.

Let the position vectors of three points lying on the plane be denoted by and respectively.

So, lies on the plane .

Therefore, is normal vector to the plane.

So, the equation of the plane passing through and normal to is

7. Find the equation of the plane passing through the following points :

and

Solution.

The equation of the plane which passes through three given points and is

Using we get the required equation of the plane

and

Solution.

The required equation of the plane is given by

8. Show that the following points are coplanar :

and

Solution.

The equation of the plane passing through the points is given by

Now, the point lies on the plane if the point satisfies the equation

So, putting on the L.H.S. of the equation we get,

Clearly, the point satisfies the equation and so the point lies on the plane.

Hence, the four points are coplanar.

and

Solution.

The equation of the plane containing three points is given by

Now, if lies on the plane , the given point must satisfy the equation

Putting on the L.H.S. of the equation we get,

Clearly, the point satisfies the equation and so the point lies on the plane.

Hence, the four points are coplanar.

9. Find the equation of the plane which makes equal (non zero) intercepts on the axes and passes through the point

Solution.

We know that the equation of the plane intercepts on the coordinate axes area and is given by

Given that the plane makes equal intercepts on the co-ordinate axes and so,

the equation of the plane can be written as

Since the plane passes through the point , we get from

Hence, the equation of the plane is