A projectile has velocity m s in some coordinate frame

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Numerical and computational mathematics

Error, Chance and the Machine, Ch Linear systems, least squares and the singular value decomposition

Problem

A projectile has velocity x=(7,24)T\mathbf{x} = (7, 24)^{\rm T} m s−1^{-1} in some coordinate frame. A single Householder reflection H=1−2vvT/(vTv)\mathsf{H} = \mathsf{1} - 2\mathbf{v}\mathbf{v}^{\rm T}/(\mathbf{v}^{\rm T}\mathbf{v}) is to map x\mathbf{x} onto the first axis (zero second component), with the sign in the construction chosen to avoid cancellation. ((a)) Find the vector v\mathbf{v} that defines the reflection. ((b)) Write down the resulting vector Hx\mathsf{H}\mathbf{x} and check its length against the original speed.

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