Paper 1, Section II, 8B

Vectors and Matrices | Part IA, 2021

(a) Consider the matrix

A=(μ112μ0μ21)A=\left(\begin{array}{rrr} \mu & 1 & 1 \\ 2 & -\mu & 0 \\ -\mu & 2 & 1 \end{array}\right)

Find the kernel of AA for each real value of the constant μ\mu. Hence find how many solutions xR3\mathbf{x} \in \mathbb{R}^{3} there are to

Ax=(112)A \mathbf{x}=\left(\begin{array}{l} 1 \\ 1 \\ 2 \end{array}\right)

depending on the value of μ\mu. [There is no need to find expressions for the solution(s).]

(b) Consider the reflection map Φ:R3R3\Phi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} defined as

Φ:xx2(xn)n\Phi: \mathbf{x} \mapsto \mathbf{x}-2(\mathbf{x} \cdot \mathbf{n}) \mathbf{n}

where n\mathbf{n} is a unit vector normal to the plane of reflection.

(i) Find the matrix HH which corresponds to the map Φ\Phi in terms of the components of n\mathbf{n}.

(ii) Prove that a reflection in a plane with unit normal n\mathbf{n} followed by a reflection in a plane with unit normal vector m\mathbf{m} (both containing the origin) is equivalent to a rotation along the line of intersection of the planes with an angle twice that between the planes.

[Hint: Choose your coordinate axes carefully.]

(iii) Briefly explain why a rotation followed by a reflection or vice-versa can never be equivalent to another rotation.

Part IA, 2021 List of Questions

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