JEE Main 2021 · 24 Feb · Shift 1 · Mathematics Q89

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MathematicsInteger+4 / -1Question 89 of 90

Let three vectors $\vec a$ , $\vec b$ and $\vec c$ be such that $\vec c$ is coplanar with $\vec a$ and $\vec b$ , $\vec a\cdot\vec c=7$ and $\vec b$ is perpendicular to $\vec c$ , where $\vec a=-\hat i+\hat j+\hat k$ and $\vec b=2\hat i+\hat k$ , then the value of $2|\vec a+\vec b+\vec c|^2$ is _______.

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Answer and worked solution

Official answer: 75

Write $\vec c=\lambda\vec a+\mu\vec b$ . $\vec b\cdot\vec c=-\lambda+5\mu=0$ and $\vec a\cdot\vec c=3\lambda-\mu=7$ give $\mu=\frac12$ , $\lambda=\frac52$ , so $\vec c=\left(-\frac32,\frac52,3\right)$ .

Sum $\vec a+\vec b+\vec c=\left(-\frac12,\frac72,5\right)$ , $|\cdot|^2=37.5$ ; doubled, 75.

Answer: 75

Topics:Dot product

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JEE Main 2021 · 24 Feb · Shift 1, Mathematics question 89