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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2023 · Mathematics · Three Dimensional Geometry

Paragraph: Let a vector a =4 i -8 j + k make angles α, β, γ with...

Paragraph: Let a vector a⃗=4i^−8j^+k^\vec{a}=4 \hat{i}-8 \hat{\stackrel{}{j}}+\hat{\stackrel{}{k}} make angles α,β,γ\alpha, \beta, \gamma with the positive directions of x,y,zx, y, z axes respectively.
Question: What is cos⁡2β+cos⁡2γ\cos 2 \beta+\cos 2 \gamma equal to?

Options

  1. A−3281-\frac{32}{81}
  2. B−1681-\frac{16}{81}
  3. C1681\frac{16}{81}
  4. D3281\frac{32}{81}

Correct answer

A. −3281-\frac{32}{81}

Step-by-step solution

cos⁡β=a2a12+a22+a32=−89cos⁡γ=a3a12+a22+a32=19cos⁡2β+cos⁡2γ=2cos⁡2β−1+2cos⁡2γ−1=2(−89)2+2(19)2−2=12881+281−2=−3281\cos \beta=\frac{a_2}{\sqrt{a_1^2+a_2^2+a_3^2}}=\frac{-8}{9}\cos \gamma=\frac{a_3}{\sqrt{a_1^2+a_2^2+a_3^2}}=\frac{1}{9}\cos 2 \beta+\cos 2 \gamma=2 \cos ^2 \beta-1+2 \cos ^2 \gamma-1=2\left(\frac{-8}{9}\right)^2+2\left(\frac{1}{9}\right)^2-2=\frac{128}{81}+\frac{2}{81}-2=\frac{-32}{81}.
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