Consider two vectors:

then the acute angle θ between two straight lines is given by:

Where b and dcan be considered as the is the direction vectors of the two lines as shown above.The lines do not have to beintersecting – the angle is the angle between them if one was moved along so they do intersect.

In case the lines L_{1} and L_{2} do not pass through the origin, we may consider that lines L_{1}’ and L_{2}’are parallel to L_{1} and L_{2} respectively and pass them throughthe origin so that these lines have the same equation with different free vector.

Suppose direction cosines of two lines L_{1} and L_{2} are given: (l_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2}) then we can find the acute angle in the following manner:

Note that, two lines containing direction ration a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are considered as:

- Perpendicular where θ =90°

Therefore:

- Parallel where θ =0°

Therefore:

Now, let us consider the angle between two lines when their equations are given in the question. If θ is assumed as an acuteangle then the angle between the lines can be written as:

Therefore, in the cartesian form we can write:

Also:

Therefore, to find the acute angle using direction ratio we can write:

Example:

Find the acute angle between the lines with vector equations:

**r** = (2**i** + **j** + **k**) + t(3**i** – 8**j** – **k**)

and **r** = (7**i** + 4**j** + **k**) + s(2**i** + 2**j** + 3**k**)

Solution:

Calculate the dot product of a and b

Now calculate the magnitude of a and b:

Substitute the values above to find the angle:

Since the answer is negative, we need to ‘make it positive’ by multiplying by -1

Example: Consider the pair of lines given below, find the angle between them.

Solution:in the cartesian form recall that:

The direction ratios of the first line can be seen from the cartesian equation are 3, 5, 4 and for the second line are 1, 1, 2. If θ is the angle between them,they can be written as:

Hence we can find the required acute angle by taking