If the projection of the vector $\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ on the sum of the two vectors $2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}$ and $-\lambda \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ is 1 , then $\lambda$ is equal to
$\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$
$\overrightarrow{\mathrm{b}}=(2-\lambda) \hat{\mathrm{i}}+6 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$
$\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{b}}|}=1, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=12-\lambda$
$(\vec{a} \cdot \vec{b})=|\vec{b}|^{2}$
$\lambda^{2}-24 \lambda+144=\lambda^{2}-4 \lambda+4+40$
$20 \lambda=100 \Rightarrow \lambda=5$
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