# NCERT Class 12-Mathematics: Chapter – 11 Three Dimensional Geometry Part 8 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 13**:

If a variable line in two adjacent positions has direction cosines *l* and , show that the small angle between the two positions is given by

**Answer**:

We have as direction cosines of a variable line in two different positions.

And

Now,

**Question 14**:

is the origin and is . Find the direction cosines of the line and the equation of plane through at right angle to .

**Answer**:

**Question 15**:

Two systems of rectangular axis have the same origin. If a plane cuts them at distances and , respectively, from the origin, prove that

**Answer**:

Consider are two system of rectangular axes.

Let their corresponding equation of plane be

And

Also, the length of perpendicular from origin to Eq. (i) and (ii) must be same.

## Long Answer (L. A)

**Question 16**:

Find the foot of perpendicular from the point to the line . Also, find the perpendicular distance from the given point to the line.

**Answer**:

**Question 17**:

Find the distance of a point from the line

**Answer**:

We have equation of the line as

Let the coordinates of L be , then Dr՚s of PL are .

Also, the direction ratios of given line are proportional to

Since, PL is perpendicular to the given line.

So, the coordinates of L are