A pyramid has a horizontal rectangular base $ABCD$, where $AB=4$ units and $AD=2$ units.

Question

A pyramid has a horizontal rectangular base $ABCD$, where $AB=4$ units and $AD=2$ units. The vertex $T$ is $5$ units vertically above the centre of the base, $O$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $AB$, $AD$ and $OT$ respectively. The point $N$ is on $TB$ such that $TN:NB=5:1$.

Find a vector equation of the line $AB$.[1]
$P$ is a point on $AB$ such that $PN$ is perpendicular to $TB$. Find the position vector of $P$.[4]
Find a cartesian equation of the plane $TBC$.[3]

A line $l$ has cartesian equation $\frac{x+2}{5}=y=\frac{z}{k}$, where $k$ is a positive real constant.

Find the exact range of values of $k$ such that $l$ is inclined at an angle of less than ${{45}^{\circ }}$ to the horizontal plane.[3]

Accepted Answer

(i) $\displaystyle \mathbf{r}=\begin{pmatrix}2\-1\0\end{pmatrix}+\lambda\begin{pmatrix}1\0\0\end{pmatrix}$, $\lambda\in\mathbb{R}$ (ii) $\displaystyle \overrightarrow{OP}=\begin{pmatrix}-\frac{1}{2}\-1\0\end{pmatrix}$ (iii) $5x+2z=10$ (iv) $0<k<\sqrt{26}$

2021 TJC Promo Q10

2021 TJC Promo Q10

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