Orthogonal vectors are independent; projections find the closest point in a subspace.
Vectors are orthogonal if their dot product is zero. The projection of b onto a is (a·b/a·a)a. Gram-Schmidt builds an orthonormal basis, and QR decomposition uses it. Orthogonal matrices preserve length.
The shadow of a vector on a floor is its projection on the floor plane.
Orthogonality makes calculations stable and simple, and projections give the best approximation when an exact solution does not exist.
Orthogonal vectors are independent; projections find the closest point in a subspace.
Orthogonality makes calculations stable and simple, and projections give the best approximation when an exact solution does not exist.
A square matrix whose columns are orthonormal, so its inverse equals its transpose and it preserves lengths and angles.
Multiply the dot product of the two vectors by the second vector, divided by its squared length, to get the component along it.
Typical applications include separating a measured signal into independent components, projecting a load onto a structural mode shape and finding the closest point on a plane to a sensor reading.
Study "Diagonalisation" first (topic 55 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Orthonormal = orthogonal unit vectors. QᵀQ = I. Projection minimises distance.
The shadow of a vector on a floor is its projection on the floor plane.
The next topic is "Least squares", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 56 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
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