(a)

\begin{align*} \text{Magnitude} & = \sqrt{3^2 + 4^2} \\ & = 5 \text{ units} \end{align*}

(b)

\begin{align*} \text{Magnitude} & = \sqrt{(-5)^2 + (12)^2} \\ & = 13 \text{ units} \end{align*}

(c)

\begin{align*} \text{Magnitude} & = \sqrt{ (-7)^2 + (-2)^2 } \\ & = 7.2801 \\ & \approx 7.28 \text{ units} \end{align*}

(d)

\begin{align*} \text{Magnitude} & = \sqrt{ (0)^2 + \left(-6{1 \over 2}\right)^2 } \\ & = 6{1 \over 2} \text{ units} \end{align*}

(e)

\begin{align*} \text{Magnitude} & = \sqrt{ 8^2 + 0^2} \\ & = 8 \text{ units} \end{align*}

(a)

$$ {-12 \choose 7}$$

(b)

$${2 \choose 0}$$

(c)

$${-4 \choose -8}$$

(d)

$${3 \choose 1.2}$$

(e)

$${0 \choose -3{1 \over 4}}$$

\begin{align*} \textbf{p} & = \textbf{q} \\ {a \choose 3} & = {-2 \choose a + 2b} \\ \\ a & = -2 \\ \\ 3 & = a + 2b \\ 3 & = (-2) + 2b \\ 3 + 2 & = 2b \\ 5 & = 2b \\ {5 \over 2} & = b \\ 2.5 & = b \end{align*}

(a)

\begin{align*} \text{Magnitude of vector} & = \sqrt{x^2 + y^2} \\ \\ |\overrightarrow{AB}| & = \sqrt{7^2 + 0^2} \\ & = 7 \text{ units} \end{align*}

(b)(i)

\begin{align*} \overrightarrow{DC} & = \overrightarrow{AB} \\ & = {7 \choose 0} \end{align*}

(b)(ii)

\begin{align*} \overrightarrow{DA} & = \overrightarrow{CB} \\ & = -\overrightarrow{BC} \\ & = - {-3 \choose 4} \\ & = {3 \choose -4} \end{align*}

\begin{align*} \overrightarrow{AB} & = {-4 \choose 3} \\ |\overrightarrow{AB}| & = \sqrt{ (-4)^2 + 3^2} \\ & = 5 \text{ units} \\ \\ \overrightarrow{CD} & = {1 \choose -2} \\ |\overrightarrow{CD}| & = \sqrt{ 1^2 + (-2)^2} \\ & = 2.236 \\ & \approx 2.24 \text{ units} \\ \\ \textbf{p} & = {3 \choose 3} \\ |\textbf{p}| & = \sqrt{3^2 + 3^2} \\ & = 4.2426 \\ & \approx 4.24 \text{ units} \\ \\ \textbf{q} & = {-2 \choose -1} \\ |\textbf{q}| & = \sqrt{(-2)^2 + (-1)^2} \\ & = 2.236 \\ & \approx 2.24 \text{ units} \\ \\ \overrightarrow{RS} & = {-2 \choose 0} \\ |\overrightarrow{RS}| & = \sqrt{(-2)^2 + 0^2} \\ & = 2 \text{ units} \\ \\ \overrightarrow{TU} & = {0 \choose 4} \\ |\overrightarrow{TU}| & = \sqrt{0^2 + 4^2} \\ & = 4 \text{ units} \end{align*}

(a)(i)

\begin{align*} \textbf{a} & = \textbf{b} \\ {x - 3 \choose 2 - y} & = {5 - x \choose y - 9} \\ \\ x - 3 & = 5 - x \\ x + x & = 5 + 3 \\ 2x & = 8 \\ x & = {8 \over 2} \\ x & = 4 \\ \\ 2 - y & = y - 9 \\ -y - y & = -9 - 2 \\ -2y & = -11 \\ 2y & = 11 \\ y & = {11 \over 2} \\ y & = 5.5 \end{align*}

(a)(ii)

\begin{align*} - \textbf{a} & = - {x - 3 \choose 2 - y} \\ & = - {(4) - 3 \choose 2 - (5.5)} \\ & = - {1 \choose -3.5} \\ & = {-1 \choose 3.5} \end{align*}

(b)(i)

\begin{align*} |\textbf{a}| & = \sqrt{(x - 3)^2 + (2 - y)^2} \\ & = \sqrt{ (x)^2 - 2(x)(3) + (3)^2 + (2)^2 - 2(2)(y) + (y)^2} \\ & = \sqrt{ x^2 - 6x + 9 + 4 - 4y + y^2 } \\ & = \sqrt{ x^2 - 6x + y^2 - 4y + 13} \\ \\ |\textbf{b}| & = \sqrt{ (5 - x)^2 + (y - 9)^2} \\ & = \sqrt{ (5)^2 - 2(5)(x) + (x)^2 + (y)^2 - 2(y)(9) + (9)^2 } \\ & = \sqrt{ 25 - 10x + x^2 + y^2 - 18y + 81} \\ & = \sqrt{ x^2 - 10x + y^2 - 18y + 106} \\ \\ |\textbf{a}| & = |\textbf{b}| \\ \sqrt{ x^2 - 6x + y^2 - 4y + 13} & = \sqrt{ x^2 - 10x + y^2 - 18y + 106} \\ x^2 - 6x + y^2 - 4y + 13 & = x^2 - 10x + y^2 - 18y + 106 \\ x^2 - x^2 - 6x + 10x + y^2 - y^2 - 4y + 18y & = 106 - 13 \\ 4x + 14y & = 93 \\ 14y & = 93 - 4x \\ y & = {93 - 4x \over 14} \end{align*}

(b)(ii)

$$ \textbf{a} \text{ and } \textbf{b} \text{ may not be in the same direction} $$

(i)

\begin{align*} |\overrightarrow{AB}| & = \sqrt{(-3)^2 + 4^2} \\ & = 5 \text{ units} \\ \\ |\overrightarrow{CD}| & = \sqrt{0^2 + 5^2} \\ & = 5 \text{ units} \\ \\ \therefore | \overrightarrow{AB} | & = |\overrightarrow{CD}| \text{ (Shown)} \end{align*}

(ii)

$$ \overrightarrow{AB} \text{ and } \overrightarrow{CD} \text{ are not in the same direction} $$

(i)

\begin{align*} \overrightarrow{AY} & = {1 \choose -4} \end{align*}

(ii)

\begin{align*} \overrightarrow{XA} & = {2 \choose 6} \\ \\ \overrightarrow{YB} & = \overrightarrow{XA} \\ & = {2 \choose 6} \end{align*}

(iii)

\begin{align*} \overrightarrow{YX} & = {-3 \choose -2} \\ \\ \overrightarrow{AC} & = \overrightarrow{YX} \\ & = {-3 \choose -2} \end{align*}

(iv)

\begin{align} \overrightarrow{AC} & = {-3 \choose -2} \\ \\ \overrightarrow{AB} & = {3 \choose 2} \\ \\ \\ \text{1) Both vectors} & \text{ have the same magnitude, } \sqrt{13} \text{ units} \\ \\ \\ \overrightarrow{AB} & = - \overrightarrow{AC} \\ \\ \text{2) } \overrightarrow{AB} \ne \overrightarrow{AC}, \phantom{0} & \text{since they are in opposite direction} \end{align}

\begin{align*} \textbf{a} & = {n \choose -3} \\ \\ |\textbf{a}| & = \sqrt{(n)^2 + (-3)^2} \\ 7 & = \sqrt{ n^2 + 9 } \\ 7^2 & = n^2 + 9 \\ 49 & = n^2 + 9 \\ 49 - 9 & = n^2 \\ 40 & = n^2 \\ \pm \sqrt{40} & = n \end{align*}

\begin{align*} \textbf{u} & = \textbf{v} \\ {13s \choose 4t} & = {6t + 20 \choose 18 - 7s} \\ \\ 13s & = 6t + 20 \phantom{000} \text{--- (1)} \\ 4t & = 18 - 7s \phantom{000} \text{--- (2)} \\ \\ \text{From (2), } 4t & = 18 - 7s \\ t & = {18 \over 4} - {7 \over 4}s \\ t & = 4.5 - 1.75s \phantom{000} \text{--- (3)} \\ \\ \text{Substitute } & \text{(3) into (1),} \\ 13s & = 6(4.5 - 1.75s) + 20 \\ 13s & = 27 - 10.5s + 20 \\ 13s + 10.5s & = 47 \\ 23.5s & = 47 \\ s & = {47 \over 23.5} \\ s & = 2 \\ \\ \text{Substitute } & s = 2 \text{ into (3),} \\ t & = 4.5 - 1.75 (2) \\ t & = 1 \\ \\ \therefore s = 2, & \phantom{.} t = 1 \end{align*}

(a)(i)

$$ \text{Rhombuses } GHIJ \text{ and } CDAB \text{ are identical} $$

(a)(ii)

$$ \overrightarrow{DC}, \overrightarrow{HG} $$

(b)(i)

$$ \overrightarrow{GD}, \overrightarrow{FE}, \overrightarrow{JA} $$

(b)(ii)

$$ \overrightarrow{GF}, \overrightarrow{KJ}, \overrightarrow{LA} $$

(b)(iii)

$$ \overrightarrow{AD}, \overrightarrow{JG}, \overrightarrow{IH} $$

(b)(iv)

$$ \overrightarrow{EG} $$

(c)

$$ \text{They have different direction} $$

(d)

$$ \text{Both vectors are in opposite direction, i.e. } \overrightarrow{BD} = - \overrightarrow{JH} $$

(e)(i)

$$ \overrightarrow{DA} $$

(e)(ii)

$$ \overrightarrow{GD} $$

(e)(iii)

$$ \overrightarrow{JK} $$