📘 vector geometry

1. **Problem Statement:** Given vectors $\overrightarrow{AB} = 3p + q$ and $\overrightarrow{BC} = 4p$, and point $O$ inside a hexagon with vertices $A, B, C, D, E, F$, find the vec

1. **Nêu bài toán:** Cho tam giác đều ABC, các điểm M, N, P sao cho \(\overrightarrow{BM} = k \overrightarrow{BC}\), \(\overrightarrow{CN} = \frac{2}{3} \overrightarrow{CA}\), \(\o

1. **State the problem:** We have parallelogram OPQT with position vectors \( \overrightarrow{OP} = \mathbf{a} \) and \( \overrightarrow{OT} = \mathbf{b} \). Point K lies on PQ suc

1. Stating the problem: We have triangle OMN with vectors $\overrightarrow{OM} = a$ and $\overrightarrow{ON} = b$. Point R lies on MN such that $MR : RN = 3 : 2$. We need to prove

1. **Problem Statement:** Find the volume of the tetrahedron determined by vectors $\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}$, $\mathbf{b} = \mathbf{i} + 2\mathbf{j} - \