Q.Find the angle between the lines whose direction ratios are $a, b, c$ and $b-c, c-a, a-b$.

Concept: Angle Between Lines — the angle $\theta$ between two lines with direction ratios $(a,b,c)$ and $(b-c, c-a, a-b)$ is given by

Step 1: Compute the numerator:

Step 2: Since the numerator is zero, $\cos\theta = 0$, so $\theta = 90^\circ$.

Concept and Intuition

The angle between two lines in space is defined as the acute angle between their direction vectors. If two lines have direction ratios $(a, b, c)$ and $(b-c, c-a, a-b)$, we are essentially comparing two vectors. The key tool is the dot product: for vectors $\vec{u}$ and $\vec{v}$,

If the dot product turns out to be zero, the lines are perpendicular — and that is exactly what happens here. The structure of the second set of ratios is cleverly designed to make the dot product vanish, regardless of the values of $a, b, c$ (as long as they are not all zero).

Step-by-Step Solution

1. Write the direction vectors.

   Let $\vec{u} = a\hat{i} + b\hat{j} + c\hat{k}$ and $\vec{v} = (b-c)\hat{i} + (c-a)\hat{j} + (a-b)\hat{k}$.

2. Compute the dot product.

3. Expand and simplify.

Every term cancels: $ab$ cancels with $-ab$, $-ac$ cancels with $+ac$, $bc$ cancels with $-bc$.

So $\vec{u} \cdot \vec{v} = 0$.

4. Interpret the result. A zero dot product means the vectors are perpendicular. Therefore, the angle between the lines is $90^\circ$.

Method: Angle Between Two Lines from Direction Ratios

Use this whenever two lines are given by their direction ratios (or direction vectors) and you must find the angle between them — including the special "are they perpendicular?" case.

Steps

Step 1: Write each line's direction ratios as a vector.

A line's position is irrelevant to the angle; only its direction matters. Call them $\vec{b_1}=(a_1,b_1,c_1)$ and $\vec{b_2}=(a_2,b_2,c_2)$.

Step 2: Form the cosine from the dot product.

The modulus in the numerator forces the acute angle, the convention for the angle between lines.

Step 3: Check the numerator first.

Compute the dot product $a_1a_2+b_1b_2+c_1c_2$ before touching the magnitudes. If it comes out $0$, the lines are perpendicular and $\theta=90^\circ$ immediately — no magnitudes needed. If it is non-zero, evaluate the two square roots and take $\theta=\cos^{-1}(\cdots)$. When the ratios are symbolic (letters, not numbers), expanding the dot product and watching for terms that cancel is exactly what reveals a hidden right angle.

Common Mistakes

Mistake 1: Trying to judge the angle by inspection instead of computing the dot product.

Why it's wrong: the second set of ratios $(b-c,\,c-a,\,a-b)$ looks unrelated to $(a,b,c)$, and guessing (e.g. "they look parallel") misses that the dot product is engineered to vanish. Correct approach: always evaluate $a(b-c)+b(c-a)+c(a-b)$; it collapses to $0$, so $\theta=90^\circ$.

Mistake 2: Wasting effort on the magnitudes before checking the numerator.

Why it's wrong: once the dot product is $0$, $\cos\theta=0$ regardless of the denominators, so computing $\sqrt{a^2+b^2+c^2}$ etc. is unnecessary. Correct approach: check the numerator first; a zero there settles the angle at once.