$$ \text { If }(\vec{a}+\vec{b}) \perp \vec{b} \text { and }(\vec{a}+2 \vec{b}) \perp \vec{a} \text {, then } $$

$$ \text { If the projection of } \vec{a}=5 \hat{\imath}+\hat{\jmath}+\lambda \hat{k} \text { on } \vec{b}=2 \hat{\imath}+6 \hat{\jmath}+3 \hat{k} \text { is } 4 \text { units, then } \lambda= $$

$$ \text { The direction ratios of the vector }(\hat{\imath}+\hat{\jmath}) \times(\hat{\jmath}+\hat{k}) \text { are } $$

Let $\vec{p}$ and $\vec{q}$ be the position vectors of P and Q with respect to the origin. If points R and S divide PQ internally and externally in the ratio 2:3 respectively, then $\overrightarrow{O R}$ and $\overrightarrow{O S}$ are perpendicular when