Question

What is the angle between the two lines having direction ratios (6, 3, 6) and (3, 3, 0)?
1. \(\dfrac{\pi}{6}\)
2. \(\dfrac{\pi}{4}\)
3. \(\dfrac{\pi}{3}\)
4. \(\dfrac{\pi}{2}\)

Answer 1

Correct Answer - Option 2 : \(\dfrac{\pi}{4}\)

CONCEPT:

The angle θ between two lines whose direction ratios are proportional to a₁, b₁, c₁ and a₂, b₂, c₂ respectively is given by: \(\cos θ = \left| {\frac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \;\sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|\)

CALCULATION:

Here, we have to find the angle between the two lines having direction ratios (6, 3, 6) and (3, 3, 0).

Here, a₁ = 6, b₁ = 3, c₁ = 6, a₂ = 3, b₂ = 3 and c₂ = 0

As we know that, if θ is the angle between two lines having direction ratios proportional to a1, b1, c1 and a2, b2, c₂ is given by: \(\cos θ = \left| {\frac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \;\sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|\)

⇒ \(\cos θ = \left| {\frac{{6 \cdot 3 + 3 \cdot 3 +6 \cdot 0}}{{\sqrt {6^2 + 3^2 + 6^2} \;\sqrt {3^2 + 3^2 + 0^2} }}} \right|\)

⇒ \(cos θ = \frac{1}{\sqrt 2}\)

⇒ θ = π/4

Hence, correct option is 2.