Angle Between Two Intersecting Lines and Shortest Distance | Mathematics (Maths) Class 12 - JEE PDF Download

Angle Between Two Intersecting Lines

If l(x₁, m₁, n₁) and l(x₂, m₂, n₂) be the direction cosines of two given lines, then the angle θ between them is given by

cos θ = l₁1₂ + m₁m₂ + n₁n₂

(i) The angle between any two diagonals of a cube is cos^-1 (1 / 3).

(ii) The angle between a diagonal of a cube and the diagonal of a face (of the cube is cos^-1(√2 / 3)

Straight Line in Space

The two equations of the line ax + by + cz + d = 0 and a’ x + b’ y + c’ z + d’ = 0 together represents a straight line.

1. Equation of a straight line passing through a fixed point A(x₁, y₁, z₁) and having direction ratios a, b, c is given by

x – x₁ / a = y – y₁ / b = z – z₁ / c, it is also called the symmetrically form of a line.

Any point P on this line may be taken as (x₁ + λa, y₁ + λb, z₁ + λc), where λ ∈ R is parameter. If a, b, c are replaced by direction cosines 1, m, n, then λ, represents distance of the point P from the fixed point A.

2. Equation of a straight line joining two fixed points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is given by

x – x₁ / x₂ – x₁ = y – y₁ / y₂ – y₁ = z – z₁ / z₂ – z₁

3. Vector equation of a line passing through a point with position vector a and parallel to vector b is r = a + λ b, where A, is a parameter.

4. Vector equation of a line passing through two given points having position vectors a and b is r = a + λ (b – a) , where λ is a parameter.

5. (a) The length of the perpendicular from a point  on the line r – a + λ b is given by

(b) The length of the perpendicular from a point P(x₁, y₁, z₁) on the line

where, 1, m, n are direction cosines of the line.

6. Skew Lines Two straight lines in space are said to be skew lines, if they are neither parallel nor intersecting.

7. Shortest Distance If l₁ and l₂ are two skew lines, then a line perpendicular to each of lines 4 and 12 is known as the line of shortest distance.

If the line of shortest distance intersects the lines l₁ and l₂ at P and Q respectively, then the distance PQ between points P and Q is known as the shortest distance between l₁ and l₂.

8. The shortest distance between the lines

9. The shortest distance between lines r = a₁ + λb₁ and r = a₂ + μb₂ is given by

10. The shortest distance parallel lines r = a₁ + λb₁ and r = a₂ + μb₂ is given by

11. Lines r = a₁ + λb₁ and r = a₂ + μb₂ are intersecting lines, if (b₁ * b₂) * (a₂ – a₁) = 0.

12. The image or reflection (x, y, z) of a point (x₁, y₁, z₁) in a plane ax + by + cz + d = 0 is given by

x – x₁ / a = y – y₁ / b = z – z₁ / c = – 2 (ax₁ + by₁ + cz₁ + d) / a² + b² + c²

13. The foot (x, y, z) of a point (x₁, y₁, z₁) in a plane ax + by + cz + d = 0 is given by

x – x₁ / a = y – y₁ / b = z – z₁ / c = – (ax₁ + by₁ + cz₁ + d) / a² + b² + c²

14. Since, x, y and z-axes pass through the origin and have direction cosines (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively. Therefore, their equations are

x – axis : x – 0 / 1 = y – 0 / 0 = z – 0 / 0

y – axis : x – 0 / 0 = y – 0 / 1 = z – 0 / 0

z – axis : x – 0 / 0 = y – 0 / 0 = z – 0 / 1