Paper 1, Section II, A

Vectors and Matrices | Part IA, 2017

(a) Define the vector product x×y\mathbf{x} \times \mathbf{y} of the vectors x\mathbf{x} and y\mathbf{y} in R3\mathbb{R}^{3}. Use suffix notation to prove that

x×(x×y)=x(xy)y(xx)\mathbf{x} \times(\mathbf{x} \times \mathbf{y})=\mathbf{x}(\mathbf{x} \cdot \mathbf{y})-\mathbf{y}(\mathbf{x} \cdot \mathbf{x})

(b) The vectors xn+1(n=0,1,2,)\mathbf{x}_{n+1}(n=0,1,2, \ldots) are defined by xn+1=λa×xn\mathbf{x}_{n+1}=\lambda \mathbf{a} \times \mathbf{x}_{n}, where a\mathbf{a} and x0\mathbf{x}_{0} are fixed vectors with a=1|\mathbf{a}|=1 and a×x00\mathbf{a} \times \mathbf{x}_{0} \neq \mathbf{0}, and λ\lambda is a positive constant.

(i) Write x2\mathbf{x}_{2} as a linear combination of a\mathbf{a} and x0\mathbf{x}_{0}. Further, for n1n \geqslant 1, express xn+2\mathbf{x}_{n+2} in terms of λ\lambda and xn\mathbf{x}_{n}. Show, for n1n \geqslant 1, that xn=λna×x0\left|\mathbf{x}_{n}\right|=\lambda^{n}\left|\mathbf{a} \times \mathbf{x}_{0}\right|.

(ii) Let XnX_{n} be the point with position vector xn(n=0,1,2,)\mathbf{x}_{n}(n=0,1,2, \ldots). Show that X1,X2,X_{1}, X_{2}, \ldots lie on a pair of straight lines.

(iii) Show that the line segment XnXn+1(n1)X_{n} X_{n+1}(n \geqslant 1) is perpendicular to Xn+1Xn+2X_{n+1} X_{n+2}. Deduce that XnXn+1X_{n} X_{n+1} is parallel to Xn+2Xn+3X_{n+2} X_{n+3}.

Show that xn0\mathbf{x}_{n} \rightarrow \mathbf{0} as nn \rightarrow \infty if λ<1\lambda<1, and give a sketch to illustrate the case λ=1\lambda=1.

(iv) The straight line through the points Xn+1X_{n+1} and Xn+2X_{n+2} makes an angle θ\theta with the straight line through the points XnX_{n} and Xn+3X_{n+3}. Find cosθ\cos \theta in terms of λ\lambda.

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