JEE Main 2022 · 28 Jun · Shift 1 · Mathematics Q24

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MathematicsInteger+4 / -1Question 24 of 90

If $\vec a=2\hat i+\hat j+3\hat k$ , $\vec b=3\hat i+3\hat j+\hat k$ and $\vec c=c_1\hat i+c_2\hat j+c_3\hat k$ are coplanar vectors and $\vec a\cdot\vec c=5$ , $\vec b\perp\vec c$ , then $122\left(c_1+c_2+c_3\right)$ is equal to ______.

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Answer and worked solution

Official answer: 150

Write $\vec c=\lambda\vec a+\mu\vec b$ . $\vec a\cdot\vec b=12$ , $|\vec a|^2=14$ , $|\vec b|^2=19$ .

Conditions. $12\lambda+19\mu=0$ and $14\lambda+12\mu=5$ give $\mu=-\frac{60}{122}$ , $\lambda=\frac{95}{122}$ .

Sum. $c_1+c_2+c_3=6\lambda+7\mu=\frac{570-420}{122}$ , so $122(c_1+c_2+c_3)=150$ .

Answer: 150

Topics:Dot product Scalar and vector triple products

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JEE Main 2022 · 28 Jun · Shift 1, Mathematics question 24