Paper 1, Section II, C

Algebra and Geometry | Part IA, 2007

Let x\mathbf{x} and y\mathbf{y} be non-zero vectors in a real vector space with scalar product denoted by xy\mathbf{x} \cdot \mathbf{y}. Prove that (xy)2(xx)(yy)(\mathbf{x} \cdot \mathbf{y})^{2} \leqslant(\mathbf{x} \cdot \mathbf{x})(\mathbf{y} \cdot \mathbf{y}), and prove also that (xy)2=(xx)(yy)(\mathbf{x} \cdot \mathbf{y})^{\mathbf{2}}=(\mathbf{x} \cdot \mathbf{x})(\mathbf{y} \cdot \mathbf{y}) if and only if x=λy\mathbf{x}=\lambda \mathbf{y} for some scalar λ\lambda.

(i) By considering suitable vectors in R3\mathbb{R}^{3}, or otherwise, prove that the inequality x2+y2+z2yz+zx+xyx^{2}+y^{2}+z^{2} \geqslant y z+z x+x y holds for any real numbers x,yx, y and zz.

(ii) By considering suitable vectors in R4\mathbb{R}^{4}, or otherwise, show that only one choice of real numbers x,y,zx, y, z satisfies 3(x2+y2+z2+4)2(yz+zx+xy)4(x+y+z)=03\left(x^{2}+y^{2}+z^{2}+4\right)-2(y z+z x+x y)-4(x+y+z)=0, and find these numbers.

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