Adding and subtracting vectors questions with answers

Questions

1. Question 1Adding column vectors

   $\mathbf{a} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -6 \\ 2 \end{pmatrix}$. Find $\mathbf{a} + \mathbf{b}$.

   Show the worked solution

   1. Step 1: Add the top numbers$-3 + (-6) = -9$
   2. Step 2: Add the bottom numbers$-4 + 2 = -2$
   3. Step 3: Write the column vector$\mathbf{a} + \mathbf{b} = \begin{pmatrix} -9 \\ -2 \end{pmatrix}$
2. Question 2Subtracting column vectors

   $\mathbf{a} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}$. Find $\mathbf{a} - \mathbf{b}$.

   Show the worked solution

   1. Step 1: Subtract the top numbers$-2 - (-4) = 2$
   2. Step 2: Subtract the bottom numbers$5 - 3 = 2$
   3. Step 3: Write the column vector$\mathbf{a} - \mathbf{b} = \begin{pmatrix} 2 \\ 2 \end{pmatrix}$
3. Question 3The triangle law

   $\overrightarrow{AB} = \begin{pmatrix} 5 \\ -4 \end{pmatrix}$ and $\overrightarrow{BC} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$. Find $\overrightarrow{AC}$.

   Show the worked solution

   1. Step 1: Go from A to C through B$\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}$
   2. Step 2: Top numbers$5 + 2 = 7$
   3. Step 3: Bottom numbers$-4 + (-2) = -6$
   4. Step 4: Write the column vector$\overrightarrow{AC} = \begin{pmatrix} 7 \\ -6 \end{pmatrix}$
4. Question 4Going back along a vector

   $\overrightarrow{AC} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}$ and $\overrightarrow{AB} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}$. Find $\overrightarrow{BC}$.

   Show the worked solution

   1. Step 1: Go from B to C through A$\overrightarrow{BC} = \overrightarrow{BA} + \overrightarrow{AC}$$\overrightarrow{BC} = \overrightarrow{AC} - \overrightarrow{AB}$
   2. Step 2: Top numbers$5 - (-2) = 7$
   3. Step 3: Bottom numbers$3 - 5 = -2$
   4. Step 4: Write the column vector$\overrightarrow{BC} = \begin{pmatrix} 7 \\ -2 \end{pmatrix}$