Let be the projection vector of , , on the vector . If , then the area of the parallelogram formed by the vectors and is _____.

16 Explanation: \begin{aligned} & =\left(\frac{\vec{b}\cdot\vec{a}}{\left|a\right|^{2}}\right)\hat{a} \\ & \Rightarrow\frac{\left(\lambda+8\right)}{9}\times\left(\hat{i}+2\hat{j}+2\hat{k}\right) \\ \mid\vec{a}+\vec{c}\mid=7 \\ & \because\left|\vec{a}\right|^{2}+\left|\vec{c}\right|^{2}+2\vec{a}\cdot\vec{c}=49 \\ & \Rightarrow9+\left(\frac{\lambda+8}{9}\right)^{2}\left(9\right)+2\left(\frac{\lambda+8}{9}\right)9=49 \\ & \Rightarrow9+\frac{\left(\lambda+8\right)^{2}}{9}+2\left(\lambda+8\right)=49 \\ & \mathrm{Let}\lambda+8=t \\ & t^{2}+81+18t=49\times9 \\ & t^{2}+18t-360=0 \\ & t^{2}+30t-12t-360=0 \\ & (t+30)(t-12)=0 \\ & t=-30,12 \\ & \lambda=-38,4 \\ & \lambda>0,\lambda=4 \\ & \vec{c}=\frac{12}{9}\left(\hat{i}+2\hat{j}+2\hat{k}\right) \\ & =\frac{4}{3}\left(\hat{i}+2\hat{j}+2\hat{k}\right) \end{aligned} \

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JEE 2025 — Mathematics PYQ

JEE | Mathematics | 2025

Let $c$ be the projection vector of $b = λ \hat{i} + 4 \hat{k}$ , $\lambda > 0$ , on the vector $a = \hat{i} + 2 \hat{j} + 2 \hat{k}$ . If $∣ a + c ∣ = 7$ , then the area of the parallelogram formed by the vectors $b$ and $c$ is _____.

JEE 2025 — Mathematics PYQ

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