The task involves solving problems related to scalar, vector, and mixed products for the given geometrical figures and points in space.

Here’s a breakdown of the tasks:

Task 1: (Given points $A(2, 1, -1)$, $B(1, 3, -3)$, $C(3, 1, 0)$, $D(7, -1, -6)$)

1. Find the external angle at vertex $A$ in triangle $ABC$:

   • This involves computing the scalar product between vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$, normalizing them, and finding the angle using: $\cos \theta = \frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{\|\overrightarrow{AB}\| \cdot \|\overrightarrow{AC}\|}$ The external angle is $180^\circ - \theta$.

2. Find the length of the median $BM$ in triangle $ABC$:

   • The median splits the opposite side into two equal halves. First, calculate the midpoint $M$ of $AC$ using: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$
   • Then find the length of $BM$ using the distance formula: $\|\overrightarrow{BM}\| = \sqrt{(x_B - x_M)^2 + (y_B - y_M)^2 + (z_B - z_M)^2}$

3. Find the area of triangle $ABC$:

   • Use the cross product of vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$. The area is: $\text{Area} = \frac{1}{2} \|\overrightarrow{AB} \times \overrightarrow{AC}\|$

4. Find the height $DH$ of tetrahedron $DABC$:

   • The height is perpendicular from $D$ to the base plane $ABC$. Use the volume formula for a tetrahedron: $V = \frac{1}{6} |\overrightarrow{DA} \cdot (\overrightarrow{AB} \times \overrightarrow{AC})|$ Then, solve for the height $DH$ using: $V = \frac{1}{3} \cdot \text{Base Area} \cdot \text{Height}$

Task 2:

Calculate the area of a parallelogram formed by vectors: $\vec{a} = 6\vec{p} - 3\vec{q}, \quad \vec{b} = 3\vec{p} + 2\vec{q}$ where $|\vec{p}| = \sqrt{3}$, $|\vec{q}| = 5$, and the angle between $\vec{p}$ and $\vec{q}$ is $\frac{2\pi}{3}$.

• First, calculate $\vec{a} \times \vec{b}$ using vector cross product properties.
• The area of the parallelogram is the magnitude of the cross product: $\text{Area} = \|\vec{a} \times \vec{b}\|$

Would you like detailed solutions and step-by-step calculations for any or all of these tasks?

Here are 5 related questions to explore further:

1. How do we calculate the angle between two vectors in 3D space?
2. What is the geometric interpretation of the cross product of two vectors?
3. How can the volume of a tetrahedron be calculated using vector operations?
4. How do the coefficients of vectors $\vec{p}$ and $\vec{q}$ affect the parallelogram's area?
5. What is the difference between an external and internal angle in geometry?

Tip: Always verify vector directions when using cross and scalar products to ensure correct results!