Dot products

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Direction and angles

2.1 Vectors as arrows

2.2 Perpendicular vectors

Applications

3.1 Comparing vectors

3.2 Finding similar text

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Scratchpads

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Making sense of dot products

I can follow the calculation. Here’s how I’ll remember it.

Note from p. 12Dot product: multiply the matching pairs, then add.

So (1, 2) · (3, 4) becomes 1 × 3 + 2 × 4 = 11.

Note from p. 18What does the dot product tell me about these vectors?

Next: draw these two vectors and compare their directions.

My explanation

A dot product gives me one number. I know how to calculate it; now I want to understand what it means.

What I’ll try next

I’ll draw two perpendicular vectors and work out their dot product. Then I’ll compare that with two vectors pointing the same way.

My prediction: changing the directions will change the result.

Editable example. Your changes stay in this demo and reset when you reload. Use the toolbar to format text. Select a linked note to remove it; Undo restores it.

Dot product: multiply the matching pairs, then add.

What does the dot product tell me about these vectors?

Draw, connect, explore — with Excalidraw.

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Your PDF stays in its original location. Margin saves your notes and scratchpads on your Mac.

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Before you get started.

Will it work with my textbooks?

Margin opens PDFs from your Mac. PDFs with selectable text work best for placing notes precisely. Scanned pages and complex layouts may offer fewer places to add notes. It supports documents up to 2,000 pages and 300 MB.

Which Macs can run Margin?

Margin runs on Apple silicon Macs (M1 or newer) with macOS 14 Sonoma or later. The current release does not support Intel Macs.

Where are my notes saved?

Your notes and scratchpads are saved locally in the Margin workspace on your Mac. Your PDF stays at its original location. Keep that file available and back up your workspace to keep your study work safe.

Do I need a subscription or an account?

No. Margin is a one-time purchase, and you don’t need a Margin account. Reading, taking notes, and using scratchpads work offline. You’ll need an internet connection to purchase and download the app.

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