The Dot Product Foundation

Linear Algebra (Matrices) · Axiom Academy

LESSON The Dot Product Foundation How two vectors of the same length collapse into a single number — the operation every matrix multiplication is built from. The dot product of and lines the two vectors up, multiplies matching components, and adds every product into one running total. Watch each pair light up and drop its product into the sum — three multiplications become a single number. The general rule: multiply componentwise, then sum Our vectors, worked out to one scalar 2. What the Number Means: Angle That scalar isn't arbitrary — it encodes the angle between the two vectors. Geometrically the dot product equals . When the vectors point the same way is large; when they're perpendicular , and the dot product vanishes. , so . The vectors reinforce each other. , so . The vectors oppose each other. is the length of 's shadow along — what the dot product measures. For and the components give 25 . Geometrically, , , and , so — the same answer. 3. The Payoff: Matrix Multiplication Here's why the dot product is the foundation. To multiply matrix A by matrix B , every entry of the result is a dot product — row i of A with column j of B . Watch row 1 pair up with column 1 to produce the top-left entry, then the rest fill in the same way. Dimensions must line up: the row length of A must equal the column length of B , because a dot product needs two equal-length vectors. A times a works — the shared 3 is the length of every row · column pair.

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