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You've provided a problem that involves calculating the dot product (producto escalar) of two vectors, \(\vec{v} = (0, \frac{1}{2})\) and \(\vec{w} = (0, -2)\), using the first definition of dot product. The dot product of two vectors \(\vec{a} = (a_1, a_2)\) and \(\vec{b} = (b_1, b_2)\) in two dimensions is calculated as follows:

\[
\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2
\]

Let's apply this formula to your vectors \(\vec{v}\) and \(\vec{w}\):

\[
\vec{v} \cdot \vec{w} = 0 \cdot 0 + \frac{1}{2} \cdot (-2)
\]

Now, I'll perform the calculation.The dot product of the vectors \(\vec{v} = (0, \frac{1}{2})\) and \(\vec{w} = (0, -2)\) is \(-1.0\).

Would you like more details or have any other questions about this calculation?

Here are some follow-up questions you might consider:

1. How does the dot product relate to the angle between two vectors?
2. What is the geometric significance of a negative dot product?
3. Can you explain how the dot product is used in physics?
4. What would be the dot product if one of the vectors were a zero vector?
5. How do you compute the dot product for three-dimensional vectors?

Tip: Remember, the dot product can give you insight into the alignment of two vectors. When the dot product is zero, the vectors are orthogonal (perpendicular to each other).

Calcula el producto escalar, usando la primera definición, de los siguientes vectores \(\vec{v} = (0, \frac{1}{2})\) y \(\vec{w} = (0, -2)\).

Learn how to calculate the dot product of two vectors \(\vec{v} = (0, \frac{1}{2})\) and \(\vec{w} = (0, -2)\). This problem covers key concepts in vector algebra and the application of the dot product formula. Suitable for students in Grades 10-12.

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