References: Dot products and projection

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Linear Algebra, Chapter 9: "Dot products and duality"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/dot-products
  Series index: https://www.3blue1brown.com/?topic=linear-algebra
  License: copyright Grant Sanderson; videos published on his site and YouTube
Clawdemy's lessons are original prose that follows the pedagogical arc of this
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Watch this next

• Dot products and duality (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. The duality argument, that dotting with a vector is the same as applying a 1-row matrix, is one of the most elegant moments in the whole series, and it lands far better watching the vector physically tip over into a transformation. The projection picture is also clearest in motion. About fourteen minutes.

Going deeper

• Essence of Linear Algebra (full series) by 3Blue1Brown. The series this track follows. The previous chapter (nonsquare matrices) set up the 1-row matrix that this lesson reveals as a vector in disguise; the next (Cross products) turns to an operation that takes two vectors and returns a third.

• Khan Academy: Linear algebra for a slower, exercise-driven treatment of dot products, projections, and angle, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track.

• Nonsquare matrices (previous lesson). The duality insight is the direct payoff of the 1-row matrix from last lesson: a transformation from n dimensions down to a single number turns out to be a vector you dot against. The two lessons are two halves of one idea.

• Cross products (next lessons). The dot product takes two vectors and returns a number that measures how much they align. The cross product, coming up, takes two vectors and returns a third vector that measures how much they spread apart (the area they span and the direction perpendicular to both). Together they are the two fundamental products of vectors.