Angle Between Two Lines

The angle between two lines in three-dimensional geometry is found by comparing their directions rather than their positions. When two lines are represented by direction ratios, direction cosines, vector form, or symmetric form, the required angle is obtained from the angle between their direction vectors.

If the direction ratios of two lines are:

First line: \[(a_1, b_1, c_1)\]

Second line: \[(a_2, b_2, c_2)\]

then the cosine of the angle \[\theta\] between them is:

If the direction cosines of the two lines are

\[(l_1, m_1, n_1)\] and \[(l_2, m_2, n_2)\], then:

If the direction ratios are \[(a_1, b_1, c_1)\] and \[(a_2, b_2, c_2)\], then:

• Perpendicular lines: \[(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0)\].

• Parallel lines: \[\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\].

• Vector form of lines: If two lines are given by:
  \[\vec{r} = \vec{a}_1 + \lambda \vec{b}_1\]
  \[\vec{r} = \vec{a}_2 + \mu \vec{b}_2\]
  then the angle between the lines is the angle between \[\vec{b}_1\] and \[\vec{b}_2\].

• Symmetric or Cartesian form
  \[\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\]

Find the angle between them.

Solution:

In vector form, the angle between the two lines is the angle between their direction vectors. Therefore, from the equations,

• \[\vec{b}_1 = \hat{i} + 2\hat{j} + 2\hat{k}\]

• \[\vec{b}_2 = 3\hat{i} + 2\hat{j} + 6\hat{k}\]

Write these as component form:

• \[\vec{b}_1 = (1, 2, 2)\]

• \[\vec{b}_2 = (3, 2, 6)\]

Now find the dot product:

• The angle between two lines depends only on their directions.

• If lines do not pass through the origin, imagine parallel lines through the origin.

• The dot-product formula is the main method for solving these questions.

• In symmetric form, denominators give direction ratios.

• Zero dot product means perpendicular lines.

• Proportional direction ratios mean parallel lines.

• The required angle is generally the acute angle.