Customizing Markdown: Interactive Vector Math

Welcome to a demonstration of how we can transform standard Markdown into a rich, textbook-quality learning experience. Instead of just static text, we use custom Astro components to create interactive notes, figures, and callouts.

Let’s explore Inner Product Spaces to see these components in action.

1. Vector Visualization

When we talk about vectors in $\mathbb{R}^2$, we often visualize them as arrows starting from the origin $(0,0)$.

Here is a visual representation of two vectors, $(1, 2)$ and $(-1, -1)$.

Notice how clearly the grid helps in identifying the components. This isn’t just an image; it’s a styled CaptionedFigure component that ensures consistent alignment and caption styling across the site.

2. Vector Addition

How do we add vectors? Geometrically, we follow the Parallelogram Law.

numerical Example

Let’s try precise addition. If we have:

• $v = (1, 2)$
• $w = (2, 1)$

Then $v + w = (1+2, 2+1) = (3, 3)$.

3. Quick Concept Review

We can also use interactive cards to summarize key concepts for the student.

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How this page is built

This entire page is written in Markdown (MDX).

• The blue boxes are <Callout> components.
• The images are <CaptionedFigure> components.
• The cards above are <ConceptCard> components.

This allows us to write educational content as easily as writing a document, while keeping the output beautiful and distinct.