(a)     Find

  (i)     $2\boldsymbol{a} + \boldsymbol{b}$ ;

  (ii)     $\left| {2\boldsymbol{a} + \boldsymbol{b}} \right|$ .

Let $2\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c} = 0$ , where $0$ is the zero vector.

(b)     Find $\boldsymbol{c}$ .

Find

  (i)     $2\boldsymbol{a} + \boldsymbol{b}$ ;

  (ii)     $\left| {2\boldsymbol{a} + \boldsymbol{b}} \right|$ .

Find $\boldsymbol{c}$ .

(a)     (i)     $2\boldsymbol{a} = \left( {\begin{array}{*{20}{c}} 4\\ { - 6} \end{array}} \right)$     (A1)

correct expression for $2\boldsymbol{a} + \boldsymbol{b}$     A1     N2

eg $\left( {\begin{array}{*{20}{c}} 5\\ { - 2} \end{array}} \right)$ , $(5, - 2)$ , $5\boldsymbol{i} - 2\boldsymbol{j}$

(ii)     correct substitution into length formula     (A1)

eg $\sqrt {{5^2} + {2^2}}$ , $\sqrt {{5^2} +  - {2^2}}$

$\left| {2\boldsymbol{a} + \boldsymbol{b}} \right| = \sqrt {29}$     A1     N2

[4 marks]

(b)     valid approach     (M1)

eg $\boldsymbol{c} = - (2\boldsymbol{a} + \boldsymbol{b})$ , $5 + x = 0$ , $- 2 + y = 0$

$\boldsymbol{c} = \left( {\begin{array}{*{20}{c}} { - 5}\\ 2 \end{array}} \right)$     A1 N2

[2 marks]

(i)     $2\boldsymbol{a} = \left( {\begin{array}{*{20}{c}} 4\\ { - 6} \end{array}} \right)$     (A1)

correct expression for $2\boldsymbol{a} + \boldsymbol{b}$     A1     N2

eg $\left( {\begin{array}{*{20}{c}} 5\\ { - 2} \end{array}} \right)$ , $(5, - 2)$ , $5\boldsymbol{i} - 2\boldsymbol{j}$

(ii)     correct substitution into length formula     (A1)

eg $\sqrt {{5^2} + {2^2}}$ , $\sqrt {{5^2} +  - {2^2}}$

$\left| {2\boldsymbol{a} + \boldsymbol{b}} \right| = \sqrt {29}$     A1     N2

[4 marks]

valid approach     (M1)

eg $\boldsymbol{c} = - (2\boldsymbol{a} + \boldsymbol{b})$ , $5 + x = 0$ , $- 2 + y = 0$

$\boldsymbol{c} = \left( {\begin{array}{*{20}{c}} { - 5}\\ 2 \end{array}} \right)$     A1 N2

[2 marks]

Most candidates comfortably applied algebraic techniques to find new vectors. However, a significant number of candidates answered part (b) as the absolute numerical value of the vector components, which suggests a misunderstanding of the modulus notation. Those who understood the notation easily made the calculation.

Most candidates comfortably applied algebraic techniques to find new vectors.

Most candidates comfortably applied algebraic techniques to find new vectors. However, a significant number of candidates answered part (b) as the absolute numerical value of the vector components, which suggests a misunderstanding of the modulus notation. Those who understood the notation easily made the calculation.