JEE Main 2022 · 26 Jun · Shift 2 · Mathematics Q16

← JEE Main 2022 · 26 Jun · Shift 2

MathematicsSingle correct+4 / -1Question 16 of 90

Let $\vec a=\hat i+\hat j+2\hat k$ , $\vec b=2\hat i-3\hat j+\hat k$ and $\vec c=\hat i-\hat j+\hat k$ be three given vectors. Let $\vec v$ be a vector in the plane of $\vec a$ and $\vec b$ whose projection on $\vec c$ is $\dfrac2{\sqrt3}$ . If $\vec v\cdot\hat j=7$ , then $\vec v\cdot\left(\hat i+\hat k\right)$ is equal to :

Answer and worked solution

Official answer: D

Write $\vec v=\lambda\vec a+\mu\vec b=(\lambda+2\mu,\ \lambda-3\mu,\ 2\lambda+\mu)$ .

Conditions. Projection on $\vec c$ : $\dfrac{2\lambda+6\mu}{\sqrt3}=\dfrac2{\sqrt3}$ , so $\lambda+3\mu=1$ ; and $\lambda-3\mu=7$ . So $\lambda=4$ , $\mu=-1$ : $\vec v=(2,7,7)$ .

Value. $\vec v\cdot(\hat i+\hat k)=9$ .

(D)

Topics:Dot product Vector operations

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JEE Main 2022 · 26 Jun · Shift 2, Mathematics question 16