# NCERT Class 12-Mathematics: Chapter – 11 Three Dimensional Geometry Part 7 (For CBSE, ICSE, IAS, NET, NRA 2023)

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**Question 6**:

Prove that the lines and are perpendicular if .

**Answer**:

We have,

And

Similarly,

From Eq. (iii) and (iv) ,

And

If these given lines are perpendicular to each other, then

Which is the required condition.

**Question 7**:

Find the equation of a plane which bisects perpendicularly the line joining the points and at right angles.

**Answer**:

**Question 8**:

Find the equation of a plane which is at a distance units from origin and the normal to which is equally inclined to coordinate axis.

**Answer**:

**Question 9**:

If the line drawn from the point meets a plane at right angle at the point , find the equation of the plane.

**Answer**:

**Question 10**:

Find the equation of the plane through the points and .

**Answer**:

**Question 11**:

Find the equations of the two lines through the origin which intersect the line at angle of each.

**Answer**:

**Question 12**:

Find the angle between the lines whose direction cosines are given by the equations .

**Answer**:

Eliminating n from both the equations, we have

Thus, Dr՚s two lines are proportional to

So, the vector parallel to these given lines are

Now,