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MathematicsSingle correct+3 / -1Question 49 of 69

IIT-JEE 2011 · Paper 1, Mathematics question 49

Let a⃗=i^+j^+k^\vec a=\hat i+\hat j+\hat k, b⃗=i^−j^+k^\vec b=\hat i-\hat j+\hat k and c⃗=i^−j^−k^\vec c=\hat i-\hat j-\hat k be three vectors. A vector v⃗\vec v in the plane of a⃗\vec a and b⃗\vec b, whose projection on c⃗\vec c is 13\dfrac1{\sqrt3}, is given by

Choose one option
Answer and worked solution

Official answer: C

In the plane of a and b: v⃗=a⃗+λb⃗=(1+λ, 1−λ, 1+λ)\vec v=\vec a+\lambda\vec b=(1+\lambda,\,1-\lambda,\,1+\lambda).

Projection on c. v⃗⋅c⃗∣c⃗∣=(1+λ)−(1−λ)−(1+λ)3=λ−13=13\dfrac{\vec v\cdot\vec c}{|\vec c|}=\dfrac{(1+\lambda)-(1-\lambda)-(1+\lambda)}{\sqrt3}=\dfrac{\lambda-1}{\sqrt3}=\dfrac1{\sqrt3}, so λ=2\lambda=2 and v⃗=3i^−j^+3k^\vec v=3\hat i-\hat j+3\hat k.

(C)

Topics:Dot productVector operations
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