Statement I If

MATHEMATICS · Subjective · A Level · Mathematics · Vectors · Vectors

Problem

Statement I If a=3i^−3j^+k^\mathbf{a} = 3\hat{i} - 3\hat{j} + \hat{k} , b=−i^+2j^+k^\mathbf{b} = -\hat{i} + 2\hat{j} + \hat{k} and c=i^+j^+k^\mathbf{c} = \hat{i} + \hat{j} + \hat{k} and d=2i^−j^\mathbf{d} = 2\hat{i} - \hat{j} , then there exist real numbers α,β,γ\alpha, \beta, \gamma such that a=αb+βc+γd\mathbf{a} = \alpha\mathbf{b} + \beta\mathbf{c} + \gamma\mathbf{d} Statement II a,b,c,d\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d} are four vectors in a 3-dimensional space. If b,c,d\mathbf{b}, \mathbf{c}, \mathbf{d} are non-coplanar, then there exist real numbers α,β,γ\alpha, \beta, \gamma such that a=αb+βc+γd\mathbf{a} = \alpha\mathbf{b} + \beta\mathbf{c} + \gamma\mathbf{d} .

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