With respect to the origin $O$, the fixed points $A$ and $B$ have position vectors \(\mathbf{a

Question

With respect to the origin $O$, the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel. The variable point $R$ has position vector $\mathbf{r} = \lambda \mathbf{a} + (1 - \lambda)\mathbf{b}$, where $\lambda$ is a real parameter. Describe geometrically the set of all possible positions of $R$.[1] It is given that the angle $AOB$ is $90^\circ$. Explain why $\mathbf{a} \cdot \mathbf{b} = 0$.[1] Among the set of all possible points $R$, the point $R^*$ is the closest to the origin $O$. Find the position vector of $R^*$. Hence state the ratio $AR^* : BR^*$ in terms of magnitudes of $\mathbf{a}$ and $\mathbf{b}$.[4]

Accepted Answer

(a) The straight line through $A$ and $B$ (b) $\mathbf a\cdot\mathbf b=0$ (c) $\overrightarrow{OR^*}=\dfrac{|\mathbf b|^2\mathbf a+|\mathbf a|^2\mathbf b}{|\mathbf a|^2+|\mathbf b|^2}$, $AR^*:BR^*=|\mathbf a|^2:|\mathbf b|^2$

2024 RI Promo Q2

2024 RI Promo Q2

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