Vector Planes (Edexcel A Level Further Maths: Core Pure): Exam Questions

$\mathbf{M}=\left(\begin{array}{ccc}2& -1& 1\\ 3& k& 4\\ 3& 2& -1\end{array}\right)$ where $k$ is a constant

Find the values of $k$ for which the matrix $\mathbf{M}$ has an inverse.

Find, in terms of $p$, the coordinates of the point where the following planes intersect

$\begin{array}{rcl}2x–y+z& =& p\\ 3x–6y+4z& =& 1\\ 3x+2y–z& =& 0\end{array}$

(i) Find the value of $q$ for which the set of simultaneous equations

$\begin{array}{rcl}2x–y+z& =& 1\\ 3x–5y+4z& =& q\\ 3x+2y–z& =& 0\end{array}$

can be solved.

(ii) For this value of $q$, interpret the solution of the set of simultaneous equations geometrically.

$\mathbf{M}=\left(\begin{array}{ccc}k& 5& 7\\ 1& 1& 1\\ 2& 1& -1\end{array}\right)$ where $k$ is a constant

Given that $k\ne 4,$ find, in terms of $k$, the inverse of the matrix $\mathbf{M}$.

Find, in terms of$p$, the coordinates of the point where the following planes intersect.

$\begin{array}{rcl}2x+5y+7z& =& 1\\ x+y+z& =& p\\ 2x+y-z& =& 2\end{array}$

(i) Find the value of $q$ for which the following planes intersect in a straight line.

$\begin{array}{rcl}4x+5y+7z& =& 1\\ x+y+z& =& q\\ 2x+y-z& =& 2\end{array}$

(ii) For this value of $q$, determine a vector equation for the line of intersection.

The line $l_1$ has equation $\frac{x-2}{4}=\frac{y-4}{-2}=\frac{z+6}{1}$

The plane $\Pi$ has equation $x-2y+z=6$

The line $l_2$ is the reflection of the line $l_1$in the plane $\Pi$

Find a vector equation of the line $l_2$

The plane ${\Pi }_1$ has vector equation

$\mathbf{r}·(3\mathbf{i}-4\mathbf{j}+2\mathbf{k})=5$

Find the perpendicular distance from the point (6, 2, 12) to the plane ${\Pi }_1$

The plane ${\Pi }_2$ has vector equation

$\mathbf{r}=\lambda (2\mathbf{i}+\mathbf{j}+5\mathbf{k})+\mu (\mathbf{i}-\mathbf{j}-2\mathbf{k})$

where $\lambda$ and $\mu$ are scalar parameters.

Show that the vector $-\mathbf{i}-3\mathbf{j}+\mathbf{k}$ is perpendicular to ${\Pi }_2$

Show that the acute angle between ${\Pi }_1$ and ${\Pi }_2$is 52° to the nearest degree.

The plane ${\Pi }_1$ has equation

$\mathbf{r}=2\mathbf{i}+4\mathbf{j}–\mathbf{k}+\lambda (\mathbf{i}+2\mathbf{j}–3\mathbf{k})+\mu (–\mathbf{i}+2\mathbf{j}+\mathbf{k})$

where $\lambda$ and $\mu$ are scalar parameters.

Find a Cartesian equation for ${\Pi }_1$

The line $l$ has equation

$\frac{x-1}{5}=\frac{y-3}{-3}=\frac{z+2}{4}$

Find the coordinates of the point of intersection $l$ with ${\Pi }_1$

The plane ${\Pi }_2$ has equation

$\mathbf{r}·(2\mathbf{i}–\mathbf{j}+3\mathbf{k})=5$

Find, to the nearest degree, the acute angle between ${\Pi }_1$ and ${\Pi }_2$

The line $l_1$ has equation

$\mathbf{r}=3\mathbf{i}+\mathbf{j}+\lambda (\mathbf{i}+\mathbf{j}+\mathbf{k})$

and the line $l_2$ has equation

$\mathbf{r}=2\mathbf{i}+p\mathbf{j}+5\mathbf{k}+\mu (\mathbf{i}+2\mathbf{j}+3\mathbf{k})$

where $\lambda$ and $\mu$ are scalar parameters and $p$ is a constant.

The plane $\Pi$ contains $l_1$ and $l_2$.

Find a vector that is perpendicular to $\Pi$.

Hence show that a Cartesian equation of $\Pi$ is

$x-2y+z=1$

Hence find the value of $p$.

Given that

• the lines $l_1$ and $l_2$ intersect at the point $A$

• the point $B$ has coordinates $(1,-4,-1)$

determine, to the nearest degree, the acute angle between $AB$ and $\Pi$.