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Deriving the Projection Formula

Handwritten conclusion of the derivation of the vector projection formula.

In this derivation, we will derive the formula for projecting one vector onto another. Vector projection is a fundamental idea in linear algebra and plays an important role in many advanced topics, including orthogonal projections, the Gram–Schmidt process, QR factorization, least-squares problems, and projection matrices.

The derivation begins with the geometric relationship between two vectors and uses the dot product and the cosine of the angle between them to obtain the projection formula.

Handwritten derivation of the vector projection formula, step 11.
Handwritten derivation of the vector projection formula, step 12.
Handwritten derivation of the vector projection formula, step 13.
Handwritten derivation of the vector projection formula, step 14.

Derivation

Handwritten derivation of the vector projection formula, step 1.
Handwritten derivation of the vector projection formula, step 2.
Handwritten derivation of the vector projection formula, step 3.
Handwritten derivation of the vector projection formula, step 4.
Handwritten derivation of the vector projection formula, step 5.
Handwritten derivation of the vector projection formula, step 6.
Handwritten derivation of the vector projection formula, step 7.
Handwritten derivation of the vector projection formula, step 8.
Handwritten derivation of the vector projection formula, step 9.
Handwritten derivation of the vector projection formula, step 10.
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Angela Morgan

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  1. Pingback: Deriving Orthogonal Vectors | The Gram-Schmidt Process – MathAngel369

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