Volume of tetrahedron using scalar triple product - Vectors: Cross Product & Distances

Edexcel FP1 Specimen Q6

11 marks Challenging +1.2

A tetrahedron has vertices $A ( 1,2,1 ) , B ( 0,1,0 ) , C ( 2,1,3 )$ and $D ( 10,5,5 )$.

Find

a Cartesian equation of the plane $A B C$.
the volume of the tetrahedron $A B C D$. The plane $\Pi$ has equation $2 x - 3 y + 3 = 0$ The point $E$ lies on the line $A C$ and the point $F$ lies on the line $A D$.
Given that $\Pi$ contains the point $B$, the point $E$ and the point $F$,
find the value of $k$ such that $\overrightarrow { A E } = k \overrightarrow { A C }$. Given that $\overrightarrow { A F } = \frac { 1 } { 9 } \overrightarrow { A D }$
show that the volume of the tetrahedron $A B C D$ is 45 times the volume of the tetrahedron $A B E F$.

Edexcel AEA 2012 June Q4

11 marks Challenging +1.8

4. $$\mathbf { a } = \left( \begin{array} { r } - 3 \\ 1 \\ 4 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5 \\ - 2 \\ 9 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 8 \\ - 4 \\ 3 \end{array} \right)$$ The points $A , B$ and $C$ with position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ ，respectively，are 3 vertices of a cube．
（a）Find the volume of the cube． The points $P , Q$ and $R$ are vertices of a second cube with $\overrightarrow { P Q } = \left( \begin{array} { l } 3 \\ 4 \\ \alpha \end{array} \right) , \overrightarrow { P R } = \left( \begin{array} { l } 7 \\ 1 \\ 0 \end{array} \right)$ and $\alpha$ a positive constant．
（b）Given that angle $Q P R = 60 ^ { \circ }$ ，find the value of $\alpha$ ．
（c）Find the length of a diagonal of the second cube．

OCR Further Additional Pure 2018 September Q4

12 marks Standard +0.3

4 The points $A , B , C$ and $P$ have coordinates ( $a , 0,0$ ), ( $0 , b , 0$ ), ( $0,0 , c$ ) and ( $a , b , c$ ) respectively, where $a , b$ and $c$ are positive constants.
The plane $\Pi$ contains $A , B$ and $C$.

(a) Use the scalar triple product to determine
- the volume of tetrahedron $O A B C$,
- the volume of tetrahedron PABC.
  (b) Hence show that the distance from $P$ to $\Pi$ is twice the distance from $O$ to $\Pi$.
- (a) Determine a vector which is normal to $\Pi$.
  (b) Hence determine, in terms of $a , b$ and $c$ only, the distance from $P$ to $\Pi$. consisting of the powers of a particular, non-singular $2 \times 2$ real matrix $\mathbf { M } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, under the operation of
  matrix multiplication. matrix multiplication.
- Explain why such a group is only possible if $\operatorname { det } ( \mathbf { M } ) = 1$ or - 1 .
- Write down values of $a , b , c$ and $d$ that would give a suitable matrix $\mathbf { M }$ for which $\mathbf { M } ^ { 6 } = \mathbf { I }$ and
Student Q observes that their class has already found a group of order 6 in a previous task; a group
Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular $2 \times 2$ real matrix $\mathbf { M } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, under the operation of
matrix multiplication. $\operatorname { det } ( \mathbf { M } ) = 1$.
Explain why such a group is only possible if $\operatorname { det } ( \mathbf { M } ) = 1$ or - 1 . Student Q believes that it is possible to construct a rational function f in the form $\mathrm { f } ( x ) = \frac { a x + b } { c x + d }$ so that the group of functions is isomorphic to the matrix group which is generated by the matrix $\mathbf { M }$ of part (iii).
(a) Write down and simplify the function f that, according to Student Q , corresponds to $\mathbf { M }$.
(b) By calculating $\mathbf { M } ^ { 3 }$, show that Student Q's suggestion does not work.
(c) Find a different function $f$ that will satisfy the requirements of the task.

Edexcel FP3 Q2

7 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{045545c7-06d9-40b6-9d01-fc792ab0aa07-01_222_241_525_2042} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $A , B$ and $C$ have position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ respectively, relative to a fixed origin $O$, as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = \mathbf { 3 i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = \mathbf { 2 i } + \mathbf { j } - \mathbf { k } .$$ Calculate

$\mathbf { b } \times \mathbf { c }$,
$\mathbf { a . } ( \mathbf { b } \times \mathbf { c } )$,
the area of triangle $O B C$,
the volume of the tetrahedron $O A B C$.