2019 MI P1 Q8
(a)
Referred to the origin $O$, the point $Q$ has position vector $\mathbf{q}$ such that
$\mathbf{q}=2\mathbf{i}-\frac{3}{2}\mathbf{j}-\frac{1}{2}\mathbf{k}$.
(i) Find the acute angle between $\mathbf{q}$ and the $y$-axis.
[2]
It is given that a vector $\mathbf{m}$ is perpendicular to the $xy$-plane and its magnitude is $1$.
(ii) With reference to the $xy$-plane, explain the geometrical meaning of $\left| \mathbf{q}\cdot \mathbf{m} \right|$ and state its value.
[2]
(b)
Referred to the origin $O$, the point $R$ has position vector $\mathbf{r}$ given by $\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}$, where $\lambda $ is a positive constant and $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors. It is known that $\mathbf{c}$is a non-zero vector that is not parallel to $\mathbf{a}$ and $\mathbf{b}$. Given that $\mathbf{c}\times \mathbf{a}=\lambda \mathbf{b}\times \mathbf{c}$, show that $\mathbf{r}$ is parallel to $\mathbf{c}$.
[2]
It is also given that $\mathbf{a}$ is a unit vector that is perpendicular to $\mathbf{b}$ and $\left| \mathbf{b} \right|=2$.
By considering $\mathbf{r}\cdot \mathbf{r}$, show that $\left| \mathbf{c} \right|=k\sqrt{4{{\lambda }^{2}}+1}$, where $k$ is a non-zero constant.
[4]
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