Edexcel A-Level Maths: Vectors & Numerical Methods

Relative to a fixed origin $O$O, the points $A$A, $B$B and $C$C have position vectors

$\vec{OA} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}, \qquad \vec{OB} = 5\mathbf{i} + 5\mathbf{j} + 9\mathbf{k}, \qquad \vec{OC} = 11\mathbf{i} + 17\mathbf{j} + 21\mathbf{k}.$OA=2i−j+3k,OB=5i+5j+9k,OC=11i+17j+21k.

(a) Find the vector $\vec{AB}$AB in the form $p\mathbf{i} + q\mathbf{j} + r\mathbf{k}$pi+qj+rk, and show that the distance $AB = 9$AB=9. (3 marks)

(b) Find a unit vector in the direction of $\vec{AB}$AB, and find the position vector of the point $P$P that divides $AB$AB in the ratio $1:2$1:2. (4 marks)

(c) Show that the points $A$A, $B$B and $C$C are collinear, and state the ratio $AB : BC$AB:BC. (5 marks)

The equation $f(x) = 0$f(x)=0, where

$f(x) = x^3 - 6x + 2,$f(x)=x3−6x+2,

has a root $\alpha$α in the interval $0 < x < 1$0<x<1.

(a) Show that there is a root of $f(x) = 0$f(x)=0 in the interval $[0, 1]$[0,1]. (2 marks)

(b) Show that the equation $f(x) = 0$f(x)=0 can be rearranged into the form $x = \dfrac{x^3 + 2}{6}$x=6x3+2​. Using the iteration formula $x_{n+1} = \dfrac{x_n^{\,3} + 2}{6}$xn+1​=6xn3​+2​ with $x_0 = 0$x0​=0, find the values of $x_1$x1​, $x_2$x2​ and $x_3$x3​, and hence state $\alpha$α correct to 3 decimal places. (4 marks)

(c) Using $x_0 = 0$x0​=0, apply the Newton-Raphson method to $f(x)$f(x) for two iterations to find an approximation to $\alpha$α. Comment on how this compares with the iteration used in part (b). (4 marks)

The vector $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}$a=2i+3j−6k.

(a) Find $|\mathbf{a}|$∣a∣ and hence write down a unit vector in the direction of $\mathbf{a}$a. (4 marks)

(b) The vector $\mathbf{b} = \mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}$b=i+cj+dk is parallel to $\mathbf{a}$a. Find the values of $c$c and $d$d, and find $|\mathbf{b}|$∣b∣. (4 marks)

The equation $x^3 + 4x - 9 = 0$x3+4x−9=0 has a single real root $\beta$β.

Taking $f(x) = x^3 + 4x - 9$f(x)=x3+4x−9 and using $x_0 = 1.5$x0​=1.5 as a first approximation, apply the Newton-Raphson method twice to find an approximation to $\beta$β, giving your answer correct to 3 decimal places.

(6 marks)

The vector $\mathbf{d} = 3\mathbf{i} - 4\mathbf{j}$d=3i−4j.

Find the two vectors of magnitude $20$20 that are parallel to $\mathbf{d}$d.

(5 marks)

The equation $x^3 - x - 6 = 0$x3−x−6=0 can be written in the form $x = \sqrt[3]{x + 6}$x=3x+6​.

Using the iteration formula $x_{n+1} = \sqrt[3]{x_n + 6}$xn+1​=3xn​+6​ with $x_0 = 1.9$x0​=1.9, find the values of $x_1$x1​ and $x_2$x2​, giving each to 4 decimal places, and state what these values suggest about the root of the equation.

(4 marks)