Class 12 NCERT Solutions – Mathematics Part ii – Chapter 11 – Three Dimensional Geometry – Miscellaneous Exercise

1. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.

Solution:

The angle θ between the lines with direction cosines a, b, c and b – c, c – a, a – b is given by:

= 90°

Thus, the required angle is 90°.

2. Find the equation of a line parallel to x-axis and passing through the origin.

Solution:

The line parallel to x-axis and passing through the origin is x- axis itself.

Let A be any point on the given line.

Thus, coordinates of A are (a,0,0) where a is any real value.

So the direction ratios of OA will be a, 0, 0.

Equation of OA will be:

Hence, the required equation is .

3. If the lines are perpendicular, find the value of k.

Solution:

Given: a₁ = 3, b₁ = 2k, c₁ = 2 and a₂ = 3k, b₂ = 1, c₂ = -5

If the lines are perpendicular, a₁a₂ + b₁b₂ + c₁c₂ = 0.

⇒ -3(3k) + 2k(1) + 2(-5) = 0

⇒ -9k + 2k -10 = 0

⇒ 7k = -10

⇒ k = -10/7

4. Find the shortest distance between lines .

Solution:

Shortest distance between two lines is given by:

Now,

Also,

Substituting these values in the formula, we have:

= 9

Thus, the shortest distance is 9 units.

5. Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines: .

Solution:

Here,

The equation of a line passing through the point (1, 2, – 4) and parallel to is given by:

Since the given two lines are perpendicular, we have:

3b₁ - 16b₂ + 7b₃ = 0

Also, 3b₁ + 8b₂ - 5b₃ = 0

Thus,

So, the direction ratios of \vec{b} are 2, 3 and 6.

Thus, equation of the vector is .

Class 12 NCERT Solutions - Chapter 11 – Three Dimensional Geometry

• Exercise 11.1
• Exercise 11.2

Related Articles:

• Angles Between two Lines in 3D Space: Solved Examples
• Equation of a Line in 3D
• Parallel and Perpendicular Lines
• Shortest Distance Between Two Lines in 3D Space | Class 12 Maths