Vectors in terms of a and b questions with answers

Questions

1. Question 1Triangles and parallelograms

   $OABC$ is a parallelogram with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{b}$. Express $\overrightarrow{OB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

   Show the worked solution

   1. Step 1: Write it using the given vectors$\overrightarrow{OB} = \mathbf{a} + \mathbf{b}$
2. Question 2Triangles and parallelograms

   In the diagram, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Express $\overrightarrow{BA}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

   Show the worked solution

   1. Step 1: Write each end from O$\overrightarrow{OB} = \mathbf{b}$$\overrightarrow{OA} = \mathbf{a}$
   2. Step 2: Go from B to A through O$\overrightarrow{BA} = \overrightarrow{BO} + \overrightarrow{OA}$$\overrightarrow{BA} = \mathbf{a} - \mathbf{b}$
3. Question 3A point dividing a side

   In the diagram, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $P$ lies on $AB$ such that $AP : PB = 3 : 2$. Express $\overrightarrow{BP}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

   Show the worked solution

   1. Step 1: Go from A to B through O$\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB}$$\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}$
   2. Step 2: PB is what fraction of AB? Use the ratio$\overrightarrow{PB} = \dfrac{2}{5}\overrightarrow{AB}$$\overrightarrow{PB} = -\dfrac{2}{5}\mathbf{a} + \dfrac{2}{5}\mathbf{b}$
   3. Step 3: BP is the opposite of PB$\overrightarrow{BP} = -\overrightarrow{PB}$$\overrightarrow{BP} = \dfrac{2}{5}\mathbf{a} - \dfrac{2}{5}\mathbf{b}$
4. Question 4A point dividing a side

   In the diagram, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $P$ lies on $AB$ such that $AP : PB = 1 : 2$. Express $\overrightarrow{PB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

   Show the worked solution

   1. Step 1: Go from A to B through O$\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB}$$\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}$
   2. Step 2: PB is what fraction of AB? Use the ratio$\overrightarrow{PB} = \dfrac{2}{3}\overrightarrow{AB}$$\overrightarrow{PB} = -\dfrac{2}{3}\mathbf{a} + \dfrac{2}{3}\mathbf{b}$
5. Question 5Points on two sides

   In the diagram, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $P$ lies on $OA$ such that $OP : PA = 3 : 1$, and $Q$ lies on $OB$ such that $OQ : QB = 1 : 1$. Express $\overrightarrow{PQ}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

   Show the worked solution

   1. Step 1: OP is what fraction of OA? Use the ratio$\overrightarrow{OP} = \dfrac{3}{4}\mathbf{a}$
   2. Step 2: OQ is what fraction of OB? Use the ratio$\overrightarrow{OQ} = \dfrac{1}{2}\mathbf{b}$
   3. Step 3: Go from P to Q through O$\overrightarrow{PQ} = \overrightarrow{PO} + \overrightarrow{OQ}$$\overrightarrow{PQ} = -\dfrac{3}{4}\mathbf{a} + \dfrac{1}{2}\mathbf{b}$
6. Question 6Points on two sides

   In the diagram, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $P$ lies on $OA$ such that $OP : PA = 2 : 3$, and $Q$ lies on $OB$ such that $OQ : QB = 3 : 2$. Express $\overrightarrow{PQ}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

   Show the worked solution

   1. Step 1: OP is what fraction of OA? Use the ratio$\overrightarrow{OP} = \dfrac{2}{5}\mathbf{a}$
   2. Step 2: OQ is what fraction of OB? Use the ratio$\overrightarrow{OQ} = \dfrac{3}{5}\mathbf{b}$
   3. Step 3: Go from P to Q through O$\overrightarrow{PQ} = \overrightarrow{PO} + \overrightarrow{OQ}$$\overrightarrow{PQ} = -\dfrac{2}{5}\mathbf{a} + \dfrac{3}{5}\mathbf{b}$