IIT-JEE 2011 · Paper 2 · Mathematics Q57

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MathematicsInteger+4Question 57 of 60

Let $\vec a=-\hat i-\hat k$ , $\vec b=-\hat i+\hat j$ and $\vec c=\hat i+2\hat j+3\hat k$ be three given vectors. If $\vec r$ is a vector such that $\vec r\times\vec b=\vec c\times\vec b$ and $\vec r\cdot\vec a=0$ , then the value of $\vec r\cdot\vec b$ is

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Answer and worked solution

Official answer: 9

Cross-product condition. $(\vec r-\vec c)\times\vec b=\vec0$ , so $\vec r=\vec c+\lambda\vec b$ .

Dot with $\vec a$ . $0=\vec c\cdot\vec a+\lambda\,\vec b\cdot\vec a=(-1-3)+\lambda(1)$ , so $\lambda=4$ .

Value. $\vec r\cdot\vec b=\vec c\cdot\vec b+4|\vec b|^2=(-1+2)+4(2)=9$ .

Answer: 9

Topics:Cross product Dot product

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IIT-JEE 2011 · Paper 2, Mathematics question 57