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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2021 · Mathematics · Three Dimensional Geometry

What is the angle between the two lines having direction ratios...

What is the angle between the two lines having direction ratios (6,3,6)(6,3,6) and (3,3,0)(3,3,0) ?

Options

  1. Aπ6\frac{\pi}{6}
  2. Bπ4\frac{\pi}{4}
  3. Cπ3\frac{\pi}{3}
  4. Dπ2\frac{\pi}{2}

Correct answer

B. π4\frac{\pi}{4}

Step-by-step solution

Angle between two lines cos⁡θ=a1a2+b1b2+c1c2a12+b12+c12⋅a22+b22+c22\cos \theta=\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}} Here, (a1,b1,c1)=(6,3,6)\left(a_1, b_1, c_1\right)=(6,3,6) and \left(a_2, b_2, c_2\right)=(, 33,0)\therefore \cos \theta=\frac{6 \times 3+3 \times 3+0}{\sqrt{6^2+3^2+6^2} \cdot \sqrt{3^2+3^2+0}}=279⋅32=12∴θ=cos⁡−1(12)=π4\begin{aligned} & =\frac{27}{9 \cdot 3 \sqrt{2}}=\frac{1}{\sqrt{2}} \\ & \therefore \theta=\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4}\end{aligned}
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