Q.Find the angle between the pair of lines $\dfrac{x+3}{3} = \dfrac{y-1}{5} = \dfrac{z+3}{4}$ and $\dfrac{x+1}{1} = \dfrac{y-4}{1} = \dfrac{z-5}{2}$.

Concept: Angle Between Lines — use direction ratios from symmetric form and the dot product formula.

Step 1 – Direction ratios

First line: $(3, 5, 4)$

Second line: $(1, 1, 2)$

Step 2 – Dot product and magnitudes

Dot product: $3(1) + 5(1) + 4(2) = 3 + 5 + 8 = 16$

Magnitudes:

$\sqrt{3^2 + 5^2 + 4^2} = \sqrt{9 + 25 + 16} = \sqrt{50} = 5\sqrt{2}$

$\sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6}$

Step 3 – Cosine of angle …

The key idea: two lines in 3D are defined by their direction vectors. The angle between the lines is simply the angle between these vectors — found using the dot product formula. There’s no need to worry about where the lines are located; only their direction matters.

1. Extract the direction vectors

   Each line is given in symmetric form $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$, where $(a,b,c)$ is the direction vector.

   For the first line: $\frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4}$ → direction vector $\vec{d}_1 = (3,5,4)$.

   For the second line: $\frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2}$ → direction vector $\vec{d}_2 = (1,1,2)$.

2. Recall the formula for the angle between two vectors

   If $\theta$ is the angle between $\vec{d}_1$ and $\vec{d}_2$, then

3. Compute the dot product

4. Compute the magnitudes

5. Find $\cos\theta$ …

Method: Angle Between Two Lines Given in Cartesian (Symmetric) Form

Use this when the lines are given as $\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$ and you need the angle between them.

Steps

Step 1: Read the direction ratios off the denominators.

In symmetric form the three denominators are the direction ratios $(a,b,c)$; the numerators (which contain the point) play no part in the angle.

Step 2: Use the direction-ratio angle formula.

$\cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}$ …

Common Mistakes

Mistake 1: Using the point coordinates instead of the denominators as direction ratios.

Why it's wrong: in $\frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4}$ the direction ratios are the denominators $(3,5,4)$, not the numbers in the numerators. Correct approach: read directions off the denominators, $(3,5,4)$ and $(1,1,2)$.

Mistake 2: Leaving the surd unsimplified or mis-simplifying it. …