# NCERT Class 12-Math՚S: Chapter – 11 Three Dimensional Geometry Part 6 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 22**:

If a line makes an angle of with each of and axis, then the angle which it makes with axis is ________

**Answer**:

Let it makes angle with axis. Then which after simplification gives

## State Whether the Following Statements Are True or False in Examples 23 and 24

**Question 23**:

The points and are collinear.

**Answer**:

Let A, B, C be the points , and , respectively. Then, the direction ratios of each of the lines AB and BC are proportional to . Therefore, the statement is true.

**Question 24**:

The vector equation of the line passing through the points and is

**Answer**:

The position vector of the points and are , and therefore, the required equation of the line is given by

Hence, the statement is true.

## 11.3 EXERCISE

### Short Answer (S. A)

**Question 1**:

Find the position vector of a point A in space such that is inclined at to OX and at to OY and units.

**Answer**:

**Question 2**:

Find the vector equation of the line which is parallel to the vector and which passes through the point

**Answer**:

**Question 3**:

Show that the lines intersect. Also, find their point of intersection.

**Answer**:

**Question 4**:

Find the angle between the lines

**Answer**:

**Question 5**:

Prove that the line through and intersects the line through and .

**Answer**:

We know that, the Cartesian equation of a line that passes through two points is

Hence, the Cartesian equation of line passes through is

And Cartesian equation of the line passes through is

If the lines intersect, then shortest distance between then should be zero.

Shortest distance between the lines

So, the given lines intersect.