Angle Between Two lines

Cartesian Form:

be two lines then angle between two lines is

Note:- If a₁,b₁,c₁, and a₂,b₂,c₂, are the direction ratios of two lines then

     i) the lines are parallel if 

     ii) the lines are perpendicular if  

Skew lines: Skew lines are lines in space which are neither parallel nor

intersecting. They lie in different planes.

Angle between two skew lines: Angle between skew lines is the angle between

two intersecting lines drawn from any point (preferably through the origin)

parallel to each of the skew lines.

Example: Find the angle between the pair of lines given by:

(x + 3)/3 = (y – 1)/5 = (z + 3)/4 and (x + 1)/1 = (y – 4)/1 = (z – 5)/2

Answer: We can solve this problem by finding the cosine of the angle between the two lines and then taking an inverse of the cosine. Let q be the angle between the two lines. We know from the formula that:

Coplanarity of Two Lines:

Coplanar lines are the lines that lie on the same plane. Prove that two lines are coplanar using the condition in vector form and Cartesian form. 

Condition for Coplanarity in Vector Form

In vector form, let us consider the equations of two straight lines to be as under:

What do these equations mean? It means that the first line passes through a point, say L, whose position vector is given by  and is parallel to . Similarly, the second line is said to pass through another point whose position vector is given by  and is parallel to

The condition for coplanarity is that the line joining the two points must be perpendicular to the product of the two vectors, m₁ and m₂. To illustrate this, we know that the line joining the two said points can be written in vector form as . So, we have:

Condition for Coplanarity in Cartesian Form

The derivation of the condition for coplanarity in Cartesian form stems from the vector form. Let us consider two points L (x₁, y₁, z₁) & M (x₂, y₂, z₂) in the Cartesian plane. Let there be two vectors  and . Their direction ratios are given by a₁, b₁, c₁ and a₂, b₂, c₂ respectively.

The vector equation of the line joining L and M can be given as:

Now use the above condition in vector form to derive our condition in Cartesian form.

Question : Are the lines (x + 3)/3 = (y – 1)/1 = (z – 5)/5 and (x + 1)/ -1 = (y – 2)/2 = (z – 5)/5 coplanar?

Answer: Comparing the equations with the general form, we have:

(x₁, y₁, z₁) = (-3, 1, 5) and (x₂, y₂, z₂) = (-1, 2, 5).

Note that a₁, b₁, c₁ = -3, 1, 5 and a₂, b₂, c₂ = -1, 2, 5.

So, by Cartesian form, we must solve the matrix:

= 2(5 – 10) – 1(-15 + 5) + 0(-6 + 1) = -10 + 10 = 0

Since the solution of the matrix gives us 0, we can say that the given lines are coplanar

This can be used as is for calculation purposes. By the condition above, the two lines would be coplanar if  LM. (m₁ x m₂) = 0. Therefore, in Cartesian form, the matrix representing this equation is given as 0.