Question

Find the angle at which the given lines are inclined to each other: \[ l_1: \frac{x - 5}{2} = \frac{y + 3}{1} = \frac{z - 1}{-3} \] \[ l_2: \frac{x}{3} = \frac{y - 1}{2} = \frac{z + 5}{-1} \]

Hint:

The angle between two lines in space can be calculated using the dot product formula: $\cos \theta = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|}$, where $\vec{d_1}$ and $\vec{d_2}$ are the direction ratios of the lines.

Answer 1

We are given two lines in symmetric form. To find the angle between two lines, we need the direction ratios of the lines. The direction ratios of line $l_1$ are obtained from the coefficients of $x$, $y$, and $z$ in the symmetric equations. Thus, the direction ratios of line $l_1$ are: \[ \vec{d_1} = \langle 2, 1, -3 \rangle \] Similarly, for line $l_2$, the direction ratios are: \[ \vec{d_2} = \langle 3, 2, -1 \rangle \] The formula for the angle $\theta$ between two lines is given by: \[ \cos \theta = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|} \] First, calculate the dot product $\vec{d_1} \cdot \vec{d_2}$: \[ \vec{d_1} \cdot \vec{d_2} = 2 \times 3 + 1 \times 2 + (-3) \times (-1) = 6 + 2 + 3 = 11 \] Now, calculate the magnitudes $|\vec{d_1}|$ and $|\vec{d_2}|$: \[ |\vec{d_1}| = \sqrt{2^2 + 1^2 + (-3)^2} = \sqrt{4 + 1 + 9} = \sqrt{14} \] \[ |\vec{d_2}| = \sqrt{3^2 + 2^2 + (-1)^2} = \sqrt{9 + 4 + 1} = \sqrt{14} \] Thus: \[ \cos \theta = \frac{11}{\sqrt{14} \times \sqrt{14}} = \frac{11}{14} \] Now find the angle $\theta$: \[ \theta = \cos^{-1} \left( \frac{11}{14} \right) \] Using a calculator: \[ \theta \approx 45.57^\circ \]