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Indicate three vectors in the plane such that any vector of this plane can be expanded in them with positive coefficients.

Prove that the point $C$ lies on the segment $A B$ if and only if there exists a number $\lambda \in [0,1]$ such that for any point $O$ the equality $\overrightarrow {O C}=\lambda \overrightarrow {O B}+(1-\lambda ) \overrightarrow {O A}$ holds. If $\lambda$ is given, in what ratio does the point $C$ divide the segment?

A regular hexagon is given, $A B C D E F,\left|A B\right|=2$. Find the coordinates of the vector $\overrightarrow {A C}$ in the basis $\overrightarrow {A B}, \overrightarrow {A D}$.

In some basis in the plane the coordinates of the vectors $\mathbf{a}(1,2), \mathbf{b}(2,3)$ and $\mathbf{c}(-1,1)$ are given. Verify that $\mathbf{a}$ and $\mathbf{b}$ are linearly independent. Find the coordinates of $\mathbf{c}$ in the basis $\mathbf{a}, \mathbf{b}$.

Three points $A, B$ and $C$ are given. Find a point $O$ such that $\overrightarrow {O A}+\overrightarrow {O B}+\overrightarrow {O C}=\mathbf{0}$. Having solved the analogous problem for four points, prove that in a triangular pyramid the segments joining the vertices with the centroids of the opposite faces intersect in one point.