Paper 1, Section II, C

Vectors and Matrices | Part IA, 2021

Using the standard formula relating products of the Levi-Civita symbol ϵijk\epsilon_{i j k} to products of the Kronecker δij\delta_{i j}, prove

a×(b×c)=(ac)b(ab)c\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}

Define the scalar triple product [a,b,c][\mathbf{a}, \mathbf{b}, \mathbf{c}] of three vectors a,b\mathbf{a}, \mathbf{b}, and c\mathbf{c} in R3\mathbb{R}^{3} in terms of the dot and cross product. Show that

[a×b,b×c,c×a]=[a,b,c]2[\mathbf{a} \times \mathbf{b}, \mathbf{b} \times \mathbf{c}, \mathbf{c} \times \mathbf{a}]=[\mathbf{a}, \mathbf{b}, \mathbf{c}]^{2}

Given a basis e1,e2,e3\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3} for R3\mathbb{R}^{3} which is not necessarily orthonormal, let

e1=e2×e3[e1,e2,e3],e2=e3×e1[e1,e2,e3],e3=e1×e2[e1,e2,e3]\mathbf{e}_{1}^{\prime}=\frac{\mathbf{e}_{2} \times \mathbf{e}_{3}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}, \quad \mathbf{e}_{2}^{\prime}=\frac{\mathbf{e}_{3} \times \mathbf{e}_{1}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}, \quad \mathbf{e}_{3}^{\prime}=\frac{\mathbf{e}_{1} \times \mathbf{e}_{2}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}

Show that e1,e2,e3\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime} is also a basis for R3\mathbb{R}^{3}. [You may assume that three linearly independent vectors in R3\mathbb{R}^{3} form a basis.]

The vectors e1,e2,e3\mathbf{e}_{1}^{\prime \prime}, \mathbf{e}_{2}^{\prime \prime}, \mathbf{e}_{3}^{\prime \prime} are constructed from e1,e2,e3\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime} in the same way that e1,e2\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, e3\mathbf{e}_{3}^{\prime} are constructed from e1,e2,e3\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}. Show that

e1=e1,e2=e2,e3=e3\mathbf{e}_{1}^{\prime \prime}=\mathbf{e}_{1}, \quad \mathbf{e}_{2}^{\prime \prime}=\mathbf{e}_{2}, \quad \mathbf{e}_{3}^{\prime \prime}=\mathbf{e}_{3}

An infinite lattice consists of all points with position vectors given by

R=n1e1+n2e2+n3e3 with n1,n2,n3Z\mathbf{R}=n_{1} \mathbf{e}_{1}+n_{2} \mathbf{e}_{2}+n_{3} \mathbf{e}_{3} \text { with } n_{1}, n_{2}, n_{3} \in \mathbb{Z}

Find all points with position vectors K\mathbf{K} such that KR\mathbf{K} \cdot \mathbf{R} is an integer for all integers n1n_{1}, n2,n3n_{2}, n_{3}.

Typos? Please submit corrections to this page on GitHub.