Vectors: RI 2024 Promo Q2

(a)[1]The set of all possible positions of $R$ is the line that passes through points $A$ and $B$, or

The set of all possible positions of $R$ is the line that passes through point $B$ (or $A$) and parallel to the vector $\overrightarrow{AB}$.

(b)[1]$\mathbf a\mathbin{\cdot}\mathbf b=|\mathbf a||\mathbf b|\cos90^\circ=0$ since $\cos90^\circ=0$.

Additional Note:

That the scalar product is 0 because 2 vectors are perpendicular (or the angle between them is $90^\circ$) is a consequence of this definition.

(c)[4]$\overrightarrow{OR^*}=\lambda\mathbf a+(1-\lambda)\mathbf b$ for some $\lambda\in\mathbb R$. $$\begin{aligned} \overrightarrow{OR^*}\mathbin{\cdot}(\mathbf a-\mathbf b)&=0,\\ [\lambda\mathbf a+(1-\lambda)\mathbf b]\mathbin{\cdot}(\mathbf a-\mathbf b)&=0,\\ \lambda|\mathbf a|^2-(1-\lambda)|\mathbf b|^2+(1-2\lambda)\mathbf a\mathbin{\cdot}\mathbf b&=0. \end{aligned}$$ Since $\mathbf a$ and $\mathbf b$ are perpendicular, $\mathbf a\mathbin{\cdot}\mathbf b=0$. $$\therefore\lambda=\frac{|\mathbf b|^2}{|\mathbf a|^2+|\mathbf b|^2}.$$ $$\overrightarrow{OR^*}=\frac{|\mathbf b|^2}{|\mathbf a|^2+|\mathbf b|^2}\mathbf a+ \frac{|\mathbf a|^2}{|\mathbf a|^2+|\mathbf b|^2}\mathbf b.$$ $$\begin{aligned} AR^*:BR^*&=1-\lambda:\lambda\\ &=\frac{|\mathbf a|^2}{|\mathbf a|^2+|\mathbf b|^2}:\frac{|\mathbf b|^2}{|\mathbf a|^2+|\mathbf b|^2}\\ &=|\mathbf a|^2:|\mathbf b|^2. \end{aligned}$$