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The lines whose vector equations are 𝑟 =2⁢̂𝑖 −3⁢̂𝑗 + 7⁢̂𝑘 + 𝜆⁢(2⁢̂𝑖+𝑝⁢̂𝑗 + 5⁢̂𝑘)⁢ and ⁢𝑟 =̂𝑖 −2⁢̂𝑗 + 3⁢𝑘 + µ⁢(3⁢̂𝑖 + 𝑝⁢̂𝑗 + 𝑝⁢̂𝑘) are perpendicular for all values of λ and µ if p =

Question

The lines whose vector equations are `r = 2hati - 3hatj + 7hatk + lambda (2hati + phatj + 5hatk) and r = hati - 2hatj + 3hatk + µ(3hati + phatj + phatk)` are perpendicular for all values of λ and µ if p =

Options

1
−1
−6
6

MCQ

Solution

Explanation:

`2hati + phatj + 5hatk and 3hati + phatj + phatk` are perpendicular

⇒ 2 × 3 + p (−p) + 5(p) = 0

⇒ p = −1 or p = 6

Hence, for p = 6, the lines are perpendicular.

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The lines whose vector equations are 𝑟 =2⁢̂𝑖 −3⁢̂𝑗 + 7⁢̂𝑘 + 𝜆⁢(2⁢̂𝑖+𝑝⁢̂𝑗 + 5⁢̂𝑘)⁢ and ⁢𝑟 =̂𝑖 −2⁢̂𝑗 + 3⁢𝑘 + µ⁢(3⁢̂𝑖 + 𝑝⁢̂𝑗 + 𝑝⁢̂𝑘) are perpendicular for all values of λ and µ if p =

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The lines whose vector equations are 𝑟 =2⁢̂𝑖 −3⁢̂𝑗 + 7⁢̂𝑘 + 𝜆⁢(2⁢̂𝑖+𝑝⁢̂𝑗 + 5⁢̂𝑘)⁢ and ⁢𝑟 =̂𝑖 −2⁢̂𝑗 + 3⁢𝑘 + µ⁢(3⁢̂𝑖 + 𝑝⁢̂𝑗 + 𝑝⁢̂𝑘) are perpendicular for all values of λ and µ if p =

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