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NDA 2025 Maths · Mathematics · Vector Algebra · easy

The vector equation of the line passing through the point with position vector 2hat i+hat j and parallel to the vector 3hat j-hat k is

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The vector equation of the line passing through the point with position vector 2i^+j^2\hat i+\hat j and parallel to the vector 3j^−k^3\hat j-\hat k is
  1. A. r⃗=(3j^−k^)+λ(2i^+j^)\vec r=(3\hat j-\hat k)+\lambda(2\hat i+\hat j)
  2. B. r⃗=(2i^+j^)⋅λ(3j^−k^)\vec r=(2\hat i+\hat j)\cdot\lambda(3\hat j-\hat k)
  3. C. r⃗=λ(2i^+j^)+μ(3j^−k^)\vec r=\lambda(2\hat i+\hat j)+\mu(3\hat j-\hat k)
  4. D. r⃗=(2i^+j^)+λ(3j^−k^)\vec r=(2\hat i+\hat j)+\lambda(3\hat j-\hat k)
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Answer: (D) r⃗=(2i^+j^)+λ(3j^−k^)\vec r=(2\hat i+\hat j)+\lambda(3\hat j-\hat k)

A line through the point a⃗\vec a parallel to b⃗\vec b is r⃗=a⃗+λb⃗\vec r=\vec a+\lambda\vec b. The first alternative interchanges the point and the direction. The second uses a dot product, which gives a scalar rather than a vector, and the last represents a whole plane through the origin, since it has two parameters.
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