The problem is in Finnish, and it deals with scalar products (dot products) in a right-angled triangle $ABC$ with a median $AD$ from the right angle. The sides $AB$ and $AC$ are given as 5 and 12 units respectively. The task is to calculate three scalar products:

• $\vec{AB} \cdot \vec{DC}$
• $\vec{BD} \cdot \vec{CA}$
• $\vec{AC} \cdot \vec{CD}$

Let's break down the problem step by step:

1. Structure of the Triangle

Since triangle $ABC$ is right-angled at point $A$, we can find the length of the hypotenuse $BC$ using the Pythagorean theorem:

$BC = \sqrt{AB^2 + AC^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

2. Midpoint $D$

Point $D$ is the midpoint of the hypotenuse $BC$. The coordinates of $D$ can be calculated as the average of the coordinates of $B$ and $C$. This will help us determine the vectors from $A$ to $D$, and subsequently compute the dot products.

3. Scalar Products

To compute the scalar products, we need to determine the vectors involved and then use the dot product formula:

$\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y$

I will now compute these scalar products.The scalar (dot) products for the given vectors in the right-angled triangle are as follows:

1. $\vec{AB} \cdot \vec{DC} = -12.5$
2. $\vec{BD} \cdot \vec{CA} = -72.0$
3. $\vec{AC} \cdot \vec{CD} = -72.0$

These values are computed using the defined vectors based on the geometry of the triangle.

Let me know if you'd like further clarification or details! Here are five related questions for further exploration:

1. How is the dot product geometrically related to the angle between two vectors?
2. Can scalar products be used to find the area of a triangle? How?
3. What is the relationship between scalar products and vector projections?
4. How does the position of the midpoint $D$ affect the scalar products?
5. What is the significance of negative dot products in geometry?

Tip: The dot product helps determine whether two vectors are perpendicular, parallel, or neither, based on the sign and value of the product.