§ 4

Suppose that in some basis the scalar product is computed by the formula $(\mathbf{a}, \mathbf{b})=\alpha_{1} \beta_{1}+\alpha_{2} \beta_{2}+\alpha_{3} \beta_{3}$. Prove that the basis is orthonormal.

Using the properties of scalar multiplication, prove that the altitudes of an arbitrary triangle intersect in one point.

Fig. 5

Draw a regular triangle $A B C$ and take the length of its side to be 1. Draw in the same figure the basis biorthogonal to the basis $\overrightarrow {A B}, \overrightarrow {A C}$.

Find the sum of the vector projections of a vector $\mathbf{a}$ onto the sides of a given regular triangle.