Question

The angle between the lines whose direction ratios are $a, b, c$ and $b - c, c - a, a - b$ is

• A. $90^\circ$
• B. $45^\circ$
• C. $30^\circ$
• D. $0^\circ$

Hint:

Whenever you see a cyclic pattern in coordinates like $(b-c, c-a, a-b)$, taking the dot product with $(a, b, c)$ will typically result in zero due to cyclic cancellation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The angle $\theta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ can be found using the dot product of their direction vectors. If the dot product is zero, the lines are perpendicular, meaning the angle between them is $90^\circ$.

Step 2: Key Formula or Approach:
Let the direction vectors of the two lines be $\vec{u}$ and $\vec{v}$. $\vec{u} = a\hat{i} + b\hat{j} + c\hat{k}$ $\vec{v} = (b - c)\hat{i} + (c - a)\hat{j} + (a - b)\hat{k}$ Calculate the dot product $\vec{u} \cdot \vec{v}$. $\vec{u} \cdot \vec{v} = a(b - c) + b(c - a) + c(a - b)$

Step 3: Detailed Explanation:
Let's expand the dot product expression: \[ \vec{u} \cdot \vec{v} = (ab - ac) + (bc - ba) + (ca - cb) \] Notice that $ab$ cancels with $-ba$, $-ac$ cancels with $ca$, and $bc$ cancels with $-cb$. \[ \vec{u} \cdot \vec{v} = ab - ac + bc - ab + ac - bc = 0 \] Since the dot product of the direction vectors is zero, the vectors are orthogonal. Therefore, the lines are perpendicular to each other. This implies that the angle $\theta$ between the lines is $90^\circ$.

Step 4: Final Answer:
The angle between the lines is $90^\circ$.