Question

Find the angle between the pair of lines \( \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4} \) and \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \).

Hint:

The direction ratios of a line given in the form \( \frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c} \) are simply the denominators \( \langle a, b, c \rangle \). Always ensure the coefficients of x, y, and z are 1 before reading the direction ratios.

Answer 1

Step 1: Understanding the Concept:
The angle between two lines in 3D space is the angle between their direction vectors. We can find the direction ratios of each line from their Cartesian equations and then use the dot product formula to find the cosine of the angle between them.
Step 2: Key Formula or Approach:
If \( \vec{b_1} = \langle a_1, b_1, c_1 \rangle \) and \( \vec{b_2} = \langle a_2, b_2, c_2 \rangle \) are the direction vectors of two lines, the angle \( \theta \) between them is given by:
\[ \cos \theta = \frac{|\vec{b_1} \cdot \vec{b_2}|}{|\vec{b_1}| |\vec{b_2}|} = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \] Step 3: Detailed Explanation or Calculation:
From the equations of the lines, we can identify their direction ratios.
For the first line, \( \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4} \), the direction vector is:
\[ \vec{b_1} = 3\hat{i} + 5\hat{j} + 4\hat{k} \] For the second line, \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \), the direction vector is:
\[ \vec{b_2} = 1\hat{i} + 1\hat{j} + 2\hat{k} \] Now, calculate the dot product \( \vec{b_1} \cdot \vec{b_2} \):
\[ \vec{b_1} \cdot \vec{b_2} = (3)(1) + (5)(1) + (4)(2) = 3 + 5 + 8 = 16 \] Next, calculate the magnitudes of the direction vectors:
\[ |\vec{b_1}| = \sqrt{3^2 + 5^2 + 4^2} = \sqrt{9 + 25 + 16} = \sqrt{50} = 5\sqrt{2} \] \[ |\vec{b_2}| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \] Now, use the formula for \( \cos\theta \):
\[ \cos\theta = \frac{16}{(5\sqrt{2})(\sqrt{6})} = \frac{16}{5\sqrt{12}} = \frac{16}{5(2\sqrt{3})} = \frac{16}{10\sqrt{3}} = \frac{8}{5\sqrt{3}} \] The angle \( \theta \) is:
\[ \theta = \cos^{-1}\left(\frac{8}{5\sqrt{3}}\right) \] Step 4: Final Answer:
The angle between the pair of lines is \( \cos^{-1}\left(\frac{8}{5\sqrt{3}}\right) \).