Discussion
For any two points \(P, Q \in \mathbb{R}^{3}\), let \(\mathit{vect}(P,Q)\) denote the vector from the point \(P\) to the point \(Q\) and let \(d(P,Q)\) denote the length of \(\mathit{vect}(P,Q)\). Let \(F: \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}\) be a linear transformation such that
\[ d(F(P), F(Q)) = d(P,Q) \]
for all \(P, Q \in \mathbb{R}^{3}\).
Which ONE of the following statements is FALSE ?
Answer: D
The FALSE statement is the one about cross products.
Because \(F\) is linear, \(\mathit{vect}(F(P),F(Q)) = F(Q) - F(P) = F(Q - P)\). So the condition says \(\|F(u)\| = \|u\|\) for every vector \(u\): \(F\) is an orthogonal map, \(F^{T}F = I\).
An orthogonal map has trivial kernel, so it is injective, and an injective linear map from \(\mathbb{R}^{3}\) to itself is also surjective. By polarisation, \(u \cdot v = \frac{1}{4}(\|u + v\|^{2} - \|u - v\|^{2})\), so preserving lengths preserves dot products. Those three statements are true.
Cross products are not preserved. In general \(F(u) \times F(v) = (\det F)\,F(u \times v)\), which is not \(u \times v\). Take \(F\) the rotation by \(\frac{\pi}{2}\) about the \(z\)-axis, so \(F(e_2) = -e_1\) and \(F(e_3) = e_3\). Then \(F(e_2) \times F(e_3) = -e_1 \times e_3 = e_2\), but \(e_2 \times e_3 = e_1\).
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