Vector product, planes and their intersections

The points $A$, $B$ and $C$ have coordinates $A(1,0,2)$, $B(3,1,1)$ and $C(2,4,0)$.

(a) Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$. [2]

(b) Find $\overrightarrow{AB}\times\overrightarrow{AC}$. [2]

(c) Hence find the area of triangle $ABC$. [2]

The plane $\Pi$ passes through the points $A(1,0,0)$, $B(0,2,0)$ and $C(0,0,3)$.

(a) Find the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$. [1]

(b) Find a normal vector to $\Pi$ and hence show that the Cartesian equation of $\Pi$ is $6x+3y+2z=6$. [4]

(c) Find the perpendicular distance from the point $P(1,1,1)$ to $\Pi$. [2]

The line $L$ has equation $\mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}$ and the plane $\Pi$ has equation $x+2y+2z=10$.

(a) Find the coordinates of the point where $L$ meets $\Pi$. [4]

(b) Find the acute angle between $L$ and $\Pi$. [3]

The vectors $\mathbf u$ and $\mathbf v$ are given by $\mathbf u=\begin{pmatrix}2\\-1\\3\end{pmatrix}$ and $\mathbf v=\begin{pmatrix}1\\3\\-2\end{pmatrix}$. The diagram shows the parallelogram with sides represented by $\mathbf u$ and $\mathbf v$.

(a) Find $\mathbf u\times\mathbf v$. [2]

(b) Verify that $\mathbf u\times\mathbf v$ is perpendicular to $\mathbf u$. [1]

(c) Find the exact area of the parallelogram. [2]

The plane $\Pi$ has vector equation $$\mathbf r=\begin{pmatrix}1\\2\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\0\\2\end{pmatrix}+\mu\begin{pmatrix}0\\1\\-1\end{pmatrix},\quad \lambda,\mu\in\mathbb R.$$

(a) Find the Cartesian equation of $\Pi$ in the form $ax+by+cz=d$. [4]

(b) The point $Q(3,\,4,\,1)$ lies in $\Pi$. Find the values of $\lambda$ and $\mu$ corresponding to $Q$. [2]

The planes $\Pi_1$ and $\Pi_2$ have equations $$\Pi_1:\ 2x-y+2z=5,\qquad \Pi_2:\ x+y-z=3.$$

(a) Find the acute angle between $\Pi_1$ and $\Pi_2$, giving your answer in degrees. [4]

The plane $\Pi_3$ has equation $x+y+pz=3$, where $p\in\mathbb R$.

(b) Given that $\Pi_3$ is perpendicular to $\Pi_1$, find the value of $p$. [2]

The planes $\Pi_1:\ x+y+z=6$ and $\Pi_2:\ 2x-y+z=3$ intersect in the line $L$.

(a) Show that the point $A(3,\,3,\,0)$ lies on both planes. [1]

(b) Find a vector equation for $L$. [4]

(c) Find the acute angle between $L$ and the plane $z=0$. [2]

Consider the system of equations $$\begin{aligned}x+2y-z&=3\\2x-y+z&=1\\x+7y+kz&=m\end{aligned}$$ where $k,m\in\mathbb R$. Each equation represents a plane.

(a) Find the value of $k$ for which the system does not have a unique solution. [4]

(b) For this value of $k$, find the value of $m$ for which the three planes meet in a line. [2]

(c) For this value of $k$ and $m\neq$ your answer to (b), describe the geometrical arrangement of the three planes, justifying your answer. [2]

The line $L$ has equation $\mathbf r=\begin{pmatrix}5\\1\\2\end{pmatrix}+t\begin{pmatrix}2\\k\\3\end{pmatrix}$ and the plane $\Pi$ has equation $x+4y-2z=-2$.

(a) Given that $L$ is parallel to $\Pi$, find the value of $k$. [2]

(b) Show that $L$ does not lie in $\Pi$. [1]

(c) Find the shortest distance between $L$ and $\Pi$. [4]

A flat mirror lies in the plane $\Pi:\ x-2y+2z=1$. A ray of light leaves the point $P(5,\,-3,\,4)$ and strikes the mirror at the point $B(1,\,0,\,0)$. The diagram shows a cross-section; $N$ is the foot of the perpendicular from $P$ to $\Pi$ and $P'$ is the reflection of $P$ in $\Pi$.

(a) Find the coordinates of $N$. [3]

(b) Hence find the coordinates of $P'$. [2]

(c) Given that the reflected ray lies on the line through $P'$ and $B$, find a vector equation for the reflected ray. [2]

(d) Find the acute angle between the incident ray $PB$ and the mirror. [2]